## (Maintained by Zhen-Qing Chen)

Probability Seminar Archive

Time:Monday, May 16, 2016 at 2:30 pm.

Location:Smith Hall 102

Speaker:Shirshendu Ganguly (UW)

Title:Scaling limit, Phase transition and Mixing of Markov Chains

Abstract:A central approach to understanding statistical mechanics is through interacting particle systems. The objective is to describe and analyze stochastic models involving spatial structure and evolution over time. Fundamental objects of interest in such systems include the equilibrium measure which the system converges to, the phenomenon of phase transition in the long term behavior and the time taken to converge to stationarity.

In this talk, we will present three examples highlighting the above aspects. We will discuss a model of competing aggregation, which exhibits conformal invariance at stationarity, a model of self organized criticality, which undergoes an absorbing state phase transition and a model of constrained Glauber dynamics, exhibiting sharp convergence to equilibrium. Several questions will also be presented.

The talk will be based on joint works with various coauthors, (articles available at, http://arxiv.org/abs/1503.06989, http://arxiv.org/abs/1508.05677 and http://arxiv.org/abs/1312.7863).

Time:Monday, May 9, 2016 at 2:30 pm.

Location:Smith Hall 102

Speaker:Daniel Ahlberg (IMPA and Uppsala university)

Title:Quenched Voronoi percolation

Abstract:In a seminal work from 1999, Benjamini, Kalai and Schramm introduced a framework for studying sensitivity of Boolean functions with respect to small portions of noise. They moreover made a series of conjectures that have been highly influential for the development since. We will discuss this development in some detail, and look more closely at the recent solution to one of these conjectures, concerning Voronoi percolation: Position a large number of points in the unit square and consider their Voronoi tessellation. Next, colour each cell either red or blue. The question is whether observing the tessellation, but not the colouring, will help us in guessing whether the colouring will produce a horizontal red crossing or not? We establish that this is not the case.

Joint work with Simon Griffiths, Rob Morris, and Vincent Tassion.

Time:Monday, May 2, 2016 at 2:30 pm.

Location:Smith Hall 102

Speaker:Ting-Kam Leonard Wong (University of Washington)

Title:Information geometry of exponentially concave functions

Abstract:A function is said to be exponentially concave if its exponential is concave. We show that exponentially concave functions on the unit simplex have remarkable connections with sequential investment and a Monge-Kantorovich optimal transport problem. Motivated by the question of optimal trading frequency, we show that each exponentially concave function induces a new geometric structure on the simplex. This structure consists of a Riemannian metric and a pair of dually coupled affine connections which define two kinds of geodesics. We show that the induced geometry is dually projectively flat. Using this geometry we prove a generalized Pythagorean theorem and construct a displacement interpolation for the associated transport problem. Our results extend the classical dually flat geometry of Bregman divergence originated from statistical inference.

Time:Monday, April 25, 2016 at 2:30 pm.

Location:Smith Hall 102

Speaker:Janos Englander (University of Colorado at Boulder)

Title:Branching Brownian motion among Poisson obstacles

Abstract:We study a branching Brownian motion $Z$ in $R^d$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide an appropriate ``skeleton" decomposition for the underlying Galton-Watson process when supercritical and show that the ``doomed" particles do not contribute to the asymptotic decay rate.

This is joint work with M. Caglar and M. Oz (Istanbul); to appear in AIHP.

Time:Tuesday, April 19, 2016 at 1:30 pm.

Location:PDL C-401

Speaker:Shayan Oveis Gharan (UW CS).

Title:Asymmetric Traveling Salesman Problem and Spectrally Thin Trees

Abstract:In an instance of the Traveling Salesman Problem (TSP) we are given a set V of n cities and their pairwise distances that satisfy the triangle inequality. The goal is to find the shortest tour that visits each city exactly once. We show that a classical algorithm due to Dantzig, Fulkerson and Johnson [DFJ54] estimates the length of shortest tour up to a poly loglog(n) factor. Our proof involves a reduction from TSP to the thin tree problem. In an instance of the thin tree spanning problem, we are given a graph G=(V,E); we want to prove the existence of a spanning tree T such that for any set S of vertices, |T(S,V-S)| << |E(S,V-S)| where T(S,V-S), E(S,V-S) correspond to the edges of T and G in the cut (S,V-S), respectively. To prove the correctness of [DJF54] algorithm we show that the graph associated with any given instance of TSP has thin spanning trees. To prove the existence of thin trees, we use many properties of random spanning tree distributions. In particular, we use the recent work of Borcea, Branden and Liggett on strongly Rayleigh probability distributions and real stable polynomials. We also exploit and extend the recent interlacing polynomial machinery of Marcus, Spielman and Srivastava.

Based on joint works with Nima Anari, Arash Asadpour, Aleksander Madry, Michel Goemans, and Amin Saberi.

Time:Monday, April 18, 2016 at 2:30 pm.

Location:Smith 102

Speaker:James Martin (Oxford University)

Title:Random graphs, forest fires and self-organised criticality

Abstract:Consider the following extension of the Erdős-Rényi random graph process; in a graph on $n$ vertices, each edge arrives at rate 1, but also each vertex is struck by lightning at rate $\lambda$, in which case all the edges in its connected component are removed. Such a "mean-field forest fire" model was introduced by Rath and Toth. For appropriate ranges of $\lambda$, the model exhibits "self-organised criticality". We investigate scaling limits, involving a multiplicative coalescent with an added "deletion" mechanism. I'll mention a few other related models, including epidemic models and "frozen percolation" processes. Joint work with Balazs Rath (Budapest).

Time:Monday, April 11, 2016 at 2:30 pm.

Location:Smith 102

Speaker:Andrey Sarantsev (University of California at Santa Barbara)

Title:PENALTY METHOD FOR OBLIQUELY REFLECTED DIFFUSIONS

Abstract:Consider a reflected Brownian motion, or, more generally, a reflected diffusion in a Euclidean domain. It can be reflected obliquely (not normally). We approximate the "solid wall" of reflection by a "soft wall" of a large drift vector field, which points inside the domain.

Time:Monday, April 4, 2016 at 2:30 pm.

Location:Smith 102

Speaker:Li-Cheng Tsai (Stanford University)

Title:Equilibrium Fluctuation of the Atlas Model

Abstract:Consider the infinite-particle Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite system of otherwise independent Brownian particles, starting at a Poisson point process on the positive half-line. We show that, at the equilibrium density of 2, the limiting fluctuations of ranked particles are characterized by the additive stochastic heat equation with the Neumann boundary condition at zero. In particular, this characterization resolves a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuations of ranked particles.

This is joint work with Amir Dembo.

Time:Monday, March 7, 2016 at 2:30 pm.

Location:SAV 155

Speaker:James Lee (University of Washington)

Title:Embeddings, martingales, and random walks on graphs

Abstract:Consider a reversible Markov chain on a discrete metric space X; the canonical example is a random walk on a graph equipped with its path metric. We consider geometry-preserving maps from X into a linear space Y (primarily a Hilbert space). One can often use such a map (or family of maps) to associate a martingale in Y to the random walk on X, and then powerful techniques from martingale theory can be brought to bear.

The martingale lens has applications in both directions: One can study metric embeddability through random walks defined on X, and also random walks via suitably constructed embeddings. I will give examples of both, with connections to geometric group theory, harmonic functions on graphs, and the non-linear theory of Banach spaces.

This is based on joint works with Yuval Peres and various other collaborators.

Time:Monday, February 29, 2016 at 2:30 pm.

Location:SAV 155

Speaker:Brent Werness (University of Washington)

Title:Probability in a Cup of Coffee

Abstract:Consider the following experiment: pour a layer of cream on top of a cup of coffee and we watch as the two fluids mix. In the beginning, the two liquids are in a simple configuration with the the layers separated. After some time passes the layers will start to mix and will exhibit a complex pattern of tendrils of cream in the coffee. Finally, the system will return to a simple state with a homogenous mix of the two liquids. This pattern of simple, to complex, and back to simple is common in physical systems, however is notoriously difficult to describe precisely in a way amenable to rigorous analysis.

,br> In this talk, I will describe joint work with Sean Carroll (Caltech), Scott Aaronson (MIT), and Varun Mohan (MIT) on a approach to this problem. After a quick discussion of the physical motivation and the concepts from theoretical computer science required, we will quickly hone in on the probabilistic problems. In particular we will give a pair of stochastic processes and prove that one of them is always simple, never producing a complex state, while the other can be shown to attain a significant level of complexity.

Time:Monday, February 22, 2016 at 2:30 pm.

Location:SAV 155

Speaker:David M. Mason (University of Delaware)

Title:The Breiman Conjecture

Abstract:Let $Y,Y_{1},Y_{2},\ldots $ be positive, nondegenerate, i.i.d.~$G$ random variables, and independently let $X,X_{1},X_{2},\ldots $ be i.i.d.~$F$ random variables. Breiman (1965) conjectured that if $X$ is nondegenerate, $E|X |<\infty $ and $\mathbb{T}_{n}=\sum X_{i}Y_{i}/\sum Y_{i}\rightarrow _{d}T$, where $T$ is nondegenerate, then necessarily $\overline{G}=1-G$ is regularly varying at infinity with index $% 0\leq \beta <1$, written $G\in D\left( \beta \right) $. In this talk we discuss the recent progress of P\'{e}ter Kevei and David Mason towards resolving this conjecture. We have shown that whenever for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, $\mathbb{T% }_{n}$ converges in distribution to a nondegenerate limit then necessarily $% G\in D\left( \beta \right) $ with $0\leq \beta <1.$ The class $\mathcal{F}$ contains the distributions of nondegenerate $X$ with a finite second moment, as well as those of $X$ in the domain of attraction of a stable law with index $1<\alpha <2 $. Our results will appear in theProceedings of the AMS. We shall also discuss the limiting distributional behavior of these self-normalized sums along subsequences and their L\'evy process analogs.

Time:Monday, February 8, 2016 at 2:30 pm.

Location:SAV 155

Speaker:Tvrtko Tadic (University of Zagreb and Microsoft Bing)

Title:Random reflections in a high dimensional tube

Abstract:Random reflections have been the subject of several recent papers. One of these contains results on reflections in two-dimensional and three-dimensional semi-infinite tubes. We will present results on the $n$-dimensional semi-infinite tubes, for $n\geq 3$, made of Lambertian material, i.e., material reflecting in the "most" random fashion. More precisely, the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. The source of light is placed far away from the end of (opening in) the tube. Using the properties of the arccosine distribution we are able to analyze the tail of the distribution of the light ray after a large number of reflections in the direction of the axis of the tube. This enables us to show that many of the results for the three-dimensional case are a special case of a more general result for the $n$-dimensional case. Further, we are able to obtain new asymptotic results on the properties of the exit distribution in dimensions $n\geq 3$ and new general results on overshoot and undershoot of a random walk. It turns out that the behavior in the two-dimensional case is different from the behavior in higher dimensions.

Joint work with Krzysztof Burdzy.

Time:Monday, February 1, 2016 at 2:30 pm.

Location:SAV 155

Speaker:Yevgeniy Kovchegov (Oregon State University)

Title:Path Coupling and Aggregate Path Coupling

Abstract:We describe and characterize an extension to the classical path coupling method referred to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. Specifically, the parameter regions for rapid mixing for the generalized Curie-Weiss-Potts model and the mean-field Blume-Capel model are derived as an application of the aggregate path coupling method.

Joint work with Peter T. Otto of Willamette University.

Time:Monday, January 11, 2016 at 2:30 pm.

Location:SIG 229

Speaker:Jun Kigami (Kyoto University, Japan)

Title:Time changes of the Brownian motion: Poincare inequality, heat kernel estimate and protodistance

Abstract:Two year's ago, I gave a talk under a similar title but it was on the case where a measure has the volume doubling property with respect to the Lebesgue measure. This time, the focus of the talk will be on non-volume doubling measures including "Liouville Brownian motion" associated with Gaussian free fields. In the case of the square, I will present a simple sufficient condition under which time change is possible, the time changed process has jointly continuous heat kernel and its satisfies some diagonal heat kernel estimate.

Time:Friday, January 8, 2016 at 2:30 pm.

Location:PDL C-401

Speaker:Maria Gordina (University of Connecticut)

Title:A random walk through sub-Riemannian geometry

Abstract:A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. In this talk, we will construct a family of geometrically natural sub-elliptic Laplacians and discuss the diffculty of defining one which is canonical. We will also construct a random walk on M which converges to a process whose infinitesimal generator is one of our sub-elliptic Laplacian operators. Examples will be provided. This is a joint work with Tom Laetsch.

Time:Monday, January 4, 2016 at 2:30 pm.

Location:SIG 229

Speaker:Tobias Johnson (University of Southern Californian)

Title:Size bias coupling and the spectral gap for random regular graphs

Abstract:The smaller the second eigenvalue of a regular graph, the stronger the expansion properties of the graph. Since this connection was discovered in the 1980s, researchers have tried to pinpoint the second eigenvalue of random regular graphs. The most prominent work in this direction was Joel Friedman's proof of Noga Alon's conjecture from 1985 that for a random d-regular graph on n vertices, the second eigenvalue is almost as small as possible, with high probability as n tends to infinity with d held fixed.

We consider the case of denser graphs, where d and n are both growing. Here, the best result (Broder, Frieze, Suen, Upfal 1999) holds only if d = o(n^(1/2)). We extend this to d = O(n^(2/3)). Our result relies on new concentration inequalities for statistics of random regular graphs based on the theory of size biased couplings, an offshoot of Stein's method. The theory we develop should be useful for proving concentration inequalities in other settings involving dependence.

This is joint work with Nicholas Cook and Larry Goldstein.

Time:Monday, December 7, 2015 at 2:30 pm.

Location:THO 325

Speaker:Avi Levy (University of Washington)

Title:FINITELY DEPENDENT GRAPH HOMOMORPHISMS

Abstract:When a child randomly paints a coloring book, adjacent regions receive distinct colors whereas distant regions remain independent. It took mathematicians until 2014 to replicate this effect, when Holroyd and Liggett discovered the first stationary $k$-dependent $q$-colorings. In this talk, I will discuss an extension of Holroyd and Liggett's construction which associates a canonical insertion procedure to every finite graph. The known colorings turn out to be diamonds in the rough: apart from multipartite analogues, they are the only $k$-dependent processes which arise from finite graphs in this manner. Time permitting, I will present extensions of these results to weighted graphs and shifts of finite type. Joint work with Alexander Holroyd.

Time:Monday, November 30, 2015 at 2:30 pm.

Location:THO 325

Speaker:Krishna B. Athreya (Iowa State University)

Title:COALESCENCE IN BRANCHING TREES AND BRANCHING RANDOM WALKS

Abstract:Consider a branching tree with one root at the origin. Assuming that at the $n$-th level there are at least two vertices. Pick two of them by simple random sampling without replacement. Now trace their lines back till they meet. Call that level $X_n$. In this talk we discuss the behavior of the distribution of $X_n$ as $n$ goes to infinity for the supercritical, critical, and subcritical Galton Watson branching trees. We also discuss the explosive case when the offspring mean is infinite and the offspring distribution is heavy tailed. We apply these results to study branching random walks. Some open problems will be described.

Time:Monday, November 23, 2015 at 2:30 pm.

Location:THO 325

Speaker:Guohuan Zhao (UW and Peking University)

Title:SOME PROPERTIES OF SUPER-BROWNIAN MOTION IN RANDOM ENVIRONMENTS

Abstract:We consider a superprocess $X=\{X_t, t\ge 0\}$ in a random environment described by a Gaussian field $W(t,x)$ whose covariance function is given by $g(x,y)(t\wedge s)$. Suppose there exists a positive function $\bar{g}$ such that $g(x,y)\leq \bar{g}(x-y)$ and the process $X$ starts from $m$, the Lebesgue measure on $\mathbb{R}^d$. We first prove that for dimension $d\geq 3$ there exists $\delta>0$ such that if $\sup_x\int_{\mathbb{R}^d} G(x,y)\bar{g}(y)dy\le\delta$ then the distribution of $X_t$ converges weakly to a non-trivial distribution $\pi^m$ as $t\to\infty$ and $\int \mu\pi^m(d\mu)=m.$ Moreover $\pi^m$ is an invariant probability distribution of $X_t$. This result implies the Conjecture 1.4 in Mytnik and Jie Xiong's paper ``Local extinction for superprocesses in random environments'' is true. We also show if $g(x,y)=g(x-y)$ with $g\in C^2(\mathbb{R}^d)$ and $g(0)$ being large enough, then $X$ suffers local extinction. The third result is: if $d=1$, then $X_t$ has compact support for all $t$ if the initial measure does.

Time:Monday, November 16, 2015 at 2:30 pm.

Location:THO 325

Speaker:Alexander E. Holroyd (Microsoft Research)

Title:RANDOM GAMES

Abstract:Alice and Bob compete in a game of skill, making moves alternately until one or other reaches a winning position, at which the game ends. Or, perhaps neither player can force a win, in which case optimal play continues forever, and we say that the game is drawn.

What is the outcome of a typical game? That is, what happens if the game itself is chosen randomly, but is known to both players, who play optimally?

I will provide some answers (any many questions) in several settings, including trees, directed and undirected lattices, and point processes. The competitive nature of game play frequently brings out some of the subtlest and most fundamental properties of probabilistic models. We will encounter continuous and discontinuous phase transitions, hard-core models, probabilistic cellular automata, bootstrap percolation, maximum matching, and stable marriage.

Based on joint works with Riddhipratim Basu, Maria Deijfen, Irene Marcovici, James Martin and Johan Wastlund.

Time:Monday, November 9, 2015 at 2:30 pm.

Location:THO 325

Speaker:Zoran Vondracek (University of Illinois, Urbana-Champaign, and University of Zagreb)

Title:BOUNDARY HARNACK PRINCIPLE AND MARTIN BOUNDARY AT INFINITY FOR FELLER PROCESSES

Abstract:A boundary Harnack principle (BHP) was recently proved for Feller processes in metric measure spaces by Bogdan, Kumagai and Kwasnicki. In this talk I will first show how their method can be modified to obtain a BHP at infinity - a result which roughly says that two non-negative function which are harmonic in an unbounded set decay at the same rate at infinity. With BHP at hand, one can identify the Martin boundary of an unbounded set at infinity with a single Martin boundary point and show that, in case infinity is accessible, this point is minimal. I will also present analogous result for a finite Martin boundary point. The local character of these results implies that minimal thinness of a set at a minimal Martin boundary point is also a local property of that set near the boundary point. Joint work with Panki Kim and Renming Song.

2015 Pacific Northwest Probability Seminar

Time:Saturday, October 24, 2015.

10:00Coffee Savery 162

11:10 - 11:50:Mathav Murugan, University of British Columbia

Title:Anomalous random walks with heavy-tailed jumps

Abstract:Sub-Gaussian estimates for the nearest neighbor random walk are typical of fractal-like graphs. In this talk, I will describe a threshold behavior of heavy-tailed random walks on such graphs. This behavior generalizes the classical threshold corresponding to the second moment condition. This talk is based on joint work with Laurent Saloff-Coste.

12:00 - 12:40:Zhenan Wang, University of Washington

Title:Stochastic De Giorgi Iteration

Abstract:We will start on the classical De Giorgi iteration for parabolic PDEs. We will explain how a stochastic version of De Giorgi iteration can be developed and applied to prove H\"older continuity for solution of stochastic partial differential equations with measurable coefficient. We will also introduce fine properties for the solutions obtained by applying the stochastic De Giorgi iterations.

12:50 - 2:20:Lunch, catered, Cascade Room in Haggett Hall, probability demos and open problems

2:30 - 3:25:Balint Virag, University of Toronto

Birnbaum Lecture, Title:: Dyson's spike and spectral measure of groups

Abstract:Consider the graph of the integers with independent random edge weights. In 1953 Dyson showed that for exactly solvable cases even a small amount of randomness results in a logarithmic spike in the spectral measure. With Marcin Kotowski, we prove that this phenomenon holds in great generality. Our result also disproves the lattice case of the Luck and Lott conjecture.

3:35 - 4:15:Juan M. Restrepo, Oregon State University

Title:: Defining a Trend of a Multi-Scale Time Series

Abstract:Defining a trend for a time series is a fundamental task in time series analyses. It has very practical outcomes: determining a trend in a financial signal, the average behavior of a dynamic process, defining exceptional and likely behavior as evidenced by a time series.

On signals and time series that have an underlying stationary statistical distribution there are a variety of ways to estimate a trend, many of which come equipped with a very concrete notion of optimality. Signals that are not statistically stationary are commonly encountered in nature, business, and the social sciences and for these the challenge of defining a trend is two-fold: computing it, and figuring out what this trend means.

Adaptive filtering is frequently explored as a means to calculating/proposing a trend. The Empirical Mode Decomposition and the Intrinsic Time Decomposition are such schemes. I will describe a practical notion of trend based upon the ITD we call a "tendency." We will briefly describe how to compute the tendency and explain its meaning.

4:20 - 4:50:Coffee break Savery 162

4:50 - 5:30:Yuval Peres, Microsoft Research

Title:: Random walk on the random graph

Abstract:: I will discuss the behavior of the random walk on two random graph models: on one hand the random regular graph with constant degree, and on the other hand the giant component of the supercritical Erdos-Renyi random graph with constant average degree. In the former case it is known that the walk mixes in logarithmic time and exhibits the cutoff phenomenon. In the latter case, while starting from the worst initial state delays mixing and precludes cutoff, it turns out that starting from a typical vertex induces the rapid mixing behavior of the regular case. (Joint work with Nathanael Berestycki, Eyal Lubetzky and Allan Sly.)

6:15No-host dinner. Restaurant:Chiang's Gourmet. 7845 Lake City Way NE, Seattle, WA 98115, Tel: (206) 527-8888

Time:Monday, October 19, 2015 at 2:30 pm.

Location:THO 325

Speaker:Anna Ben-Hamou (University of Paris-Diderot and Microsoft)

Title:CUTOFF FOR NON-BACKTRACKING RANDOM WALKS ON SPARSE RANDOM GRAPHS

Abstract:A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. In this talk, we will consider the case of non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we will establish the cutoff phenomenon, determine its precise window, and prove that the cutoff profile approaches a remarkably simple, universal shape.

This is a joint work with Justin Salez (Paris-Diderot).

Time:Monday, October 12, 2015 at 2:30 pm.

Location:THO 325

Speaker:Clayton Barnes (University of Washington)

Title:BROWNIAN MOTION AND AN INERT PARTICLE

Abstract:In his 2001 paper, Frank Knight constructed a process that moves with inertia and is impinged upon by a Brownian motion. Later, Bass, Burdzy, Chen and Hairer found the stationary distribution for the process extended to multi-dimensions. In this talk we give a discrete version of the inert particle, showing it converges in distribution after interpolation. In addition, we characterize the distribution of the maximum, and show that the sequence of local maximums forms a submartingale when a reflected boundary is introduced.

Time:Monday, October 5, 2015 at 2:30 pm.

Location:THO 325

Speaker:Konstantin Tikhomirov (University of Alberta)

Title:THE SMALLEST SINGULAR VALUE OF RANDOM MATRICES WITH INDEPENDENT ROWS

Abstract:We consider a classical problem of estimating the least singular value of random rectangular and square matrices with independent identically distributed entries as well as a more general model of random matrices with i.i.d. rows. The novelty of our results consists primarily in imposing very weak, or nonexisting, moment assumptions on the distribution of the entries. In particular, we show that the smallest singular value of a sufficiently tall $N\times n$ rectangular matrix with i.i.d. entries with certain condition on the Levy concentration function is of order $\sqrt{N}$ with a large probability.

Further, we extend a fundamental result of Bai and Yin from early 1990-es on the limiting behaviour of the smallest singular value of rectangular matrices, by dropping the assumption of bounded 4th moment (a problem whether the assumption was necessary was discussed, in particular, in a book of Bai and Silverstein).

Finally, we will discuss very recent joint results with Elizaveta Rebrova (University of Michigan) and Djalil Chafai (Paris Dauphine) regarding invertibility of random square matrices with i.i.d. heavy-tailed entries and Bai--Yin type convergence for matrices with i.i.d. log-concave rows, respectively.

To obtain the results, we have used various techniques including special versions of the $\varepsilon$-net argument and rank one update approach of Batson--Spielman--Srivastava.

Time:Monday, September 21, 2015 at 2:30 pm.

Location:THO 134

Speaker:Guenter Last (Karlsruhe Institute of Technology)

Title:INVARIANT TRANSPORTS AND MATCHINGS OF RANDOM MEASURES

Abstract:The extra head problem formulated and solved by Liggett (2001) requires finding a head in an infinite sequence of independent fair coin tosses, without changing the (joint) distribution of the rest of the sequence. Holroyd and Peres (2005) constructed a a stable and shift invariant transport (marriage) of Lebesgue measure and an ergodic point process with unit intensity. Invariant matchings of stationary point processes were studied by Holroyd, Pemantle, Peres and Schramm (2008). In this talk we shall discuss matching and embedding problems for a two-sided standard Brownian motion and explain the close relationship of all these matching problems with shift invariant transports and Palm measures of stationary random measures. This talk is based on joint work with Peter Morters (Bath) and Hermann Thorisson (Reykjavik).

Time:Monday, June 1, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Bjorn Kjos-Hanssen (University of Hawaii)

Title:Kolmogorov structure functions for automatic complexity

Abstract:We study an analogue of Kolmogorov's notion of structure function introduced in 1973, with Kolmogorov complexity replaced by Shallit and Wang's (2001) automatic complexity. We discuss the prospects for using it for model selection in statistics. We prove an upper bound which is piecewise smooth, related to the binary entropy function, and appears to be fairly sharp based on numerical evidence.

The paper is loosely coupled with the following software:

- [Complexity Guessing Game][1]

- [Complexity Option Game][2]

- [Structure Function Calculator][3]

[1]: http://math.hawaii.edu/wordpress/bjoern/software/web/complexity-guessing-game/

[2]: http://math.hawaii.edu/wordpress/bjoern/software/web/complexity-option-game/

[3]: http://math.hawaii.edu/wordpress/bjoern/structure-function-calculator/

Time:Monday, May 18, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Shirshendu Ganguly (UW)

Title:Competitive erosion is conformally invariant

Abstract:We study a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite competitive version of the celebrated Internal DLA growth model.

We establish conformal invariance of competitive erosion on discretizations of smooth planar simply connected domains. This is done by showing that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic.

The talk is based on joint work with Yuval Peres.

joint DG-PDE/Probability seminar

Time:Wednesday, May 13, 2015 at 4 pm.

Location:PDL C-36

Speaker:Carlo Morpurgo (University of Missouri at Columbia)

Title:Adams and Moser-Trudinger inequalities: recent results on spaces of infinite measure

Abstract:A Moser-Trudinger inequality is a statement about the exponential integra- bility of functions that belongs to the so-called critical Sobolev space W^{k;n/k}. Best constants in such inequalities were often crucial for the solutions of problems in conformal geometry and certain nonlinear PDEs. In 1988 D. Adams invented a very successful strategy for proving such inequalities on bounded domains of R^n, via the use of Riesz potentials, the O'Neil Lemma and other ingenious estimates. Since then many many authors used Adams' approach to derive sharp Moser-Trudinger results in various settings, like general compact Riemannian manifolds, or on bounded domains in subRiemannian manifolds. In 2011 in joint work with Luigi Fontana we formulated and proved an abstract version of Adams' result valid in arbitrary measure spaces with finite measure, that allowed us to obtain old and new results on sets of finite measure by simply checking a couple of kernel estimates.

On the other hand, optimal results on sets of infinite measure are relatively few, mostly for W^{1, n} and W^{2, n/2}, and recently there has been a flurry of work attempting to complete the existing gaps. In recent work with Fontana we estabilshed the full Adams and Moser-Trudinger results on the whole of R^n, for the Riesz potentials, the higher order gradients, and also more general elliptic operators of constant coecients. We also established results on the hyperbolic space for arbitrary order.

In this talk I will present a little bit of the motivation and background, some older results, some very recent ones.

Time:Monday, May 11, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Antonio Blanca (UC Berkeley)

Title:Dynamics for the mean-field random-cluster model

Abstract:The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this work we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (p_l,p_r) of the model parameter p in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is O(log n) when p is not in [p_l,p_r], and exp(\sqrt(n)) when p is in (p_l,p_r). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (p_l,p_r).

This is joint work with Alistair Sinclair.

Time:Monday, April 27, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Douglas Rizzolo (UW)

Title:The structure of thin Young tableaux

Abstract:In this talk we will look at the structure of standard Young tableaux with a fixed number of rows as the size of the tableaux goes to infinity. By treating the a tableaux as a collection of nonintersecting paths, we will draw connections with random matrices, conditioned Brownian motion, and pattern-avoiding permutations.

Time:Monday, April 20, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Andrey Sarantsev (UW)

Title:COMPETING BROWNIAN PARTICLES

Abstract:Consider a system of particles on the real line. Rank them from left to right: the leftmost particle has rank one, the next leftmost has rank two, etc. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. When particles collide and exchange ranks, they exchange their drift and diffusion coefficients. We prove new results for these systems, both finite and infinite.

Time:Monday, April 13, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Zhen-Qing Chen (UW)

Title:Heat kernels for non-local operators

Abstract:Transition density function encodes all the information about a Markov process. It is the fundamental solution (also called heat kernel) of the infinitesimal generator of the corresponding Markov process. When the Markov process is a diffusion on Euclidean space, its infinitesimal generator is a differential operator. When the Markov process has discontinuous sample paths, its infinitesimal generator is a non-local operator. Heat kernels for second order elliptic operators have been well studied and there are many beautiful results. The study of heat kernels for non-local operators is quite recent. In this talk, I will survey the recent progress in the study of heat kernels of non-local operators, focusing on their two-sided estimates, stability, and their applications to solutions of stochastic differential equations driven by Levy processes.

Time:Monday, April 6, 2015 at 2:30 pm.

Location:SMI 102

Speaker:Thomas Rothvoss (UW)

Title:Constructive discrepancy minimization for convex sets

Abstract:A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett-Meka algorithm finds a half integral point in any "large enough" polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set K with measure at least exp(-n/500), the following algorithm finds a point y in K \cap [-1,1]^n with Omega(n) coordinates in {-1,+1}: (1) take a random Gaussian vector x; (2) compute the point y in K \cap [-1,1]^n that is closest to x. (3) return y. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a Theorem of Gluskin and Giannopoulos.

Time:Monday, March 9, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Andrey Pilipenko (Michigan State)

Title:On the strong solutions of stochastic equations with Lévy noise.

Abstract:Stochastic differential equations with additive symmetric Levy noise is considered in a multidimensional Euclidean space. Suppose that a drift is a function of bounded variation. If a derivative of the drift (considered in the generalized sense) is a measure from a proper Kato class, we prove the existence of the strong solution to the SDE. Moreover, this solution is L_p differentiable with respect to the starting point.

Time:Monday, March 2, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Tvrkto Tadic (UW)

Title:Random reflections and stopped random walks

Abstract:We study the behavior of a random reflection model (introduced by Comets et al.) in the setting of a semi-infinite tube. It turns out that this model is governed by a special case of a stopped random walk. We try to describe the behavior of the light rays exiting the tube when the source of light is far away from the exit. Using the results obtained by Doney, Erickson and Rogozin we find many explicit estimates about the limiting behavior of a symmetric random walk, its ladder variable process, the minimal solution of the Wiener-Hopf equation and the undershoot and overshoot. This in the end produces nice results about the luminosity of the points in the tube. Joint work in progress with Krzysztof Burdzy.

Time:Monday, February 23, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Harish Ramadas (UW)

Title:Mixing of the Noisy Voter Model

Abstract:The noisy voter model is a simple `interacting particle system' that can serve as a crude stochastic model for the spread of opinions in a human population, or the competition for territory between two species. We study the convergence to equilibrium (`mixing') of this model, and show that this happens extremely fast -- more precisely, on an arbitrary graph with n vertices, the model mixes in time of O(log n), for arbitrarily small values of the `noise parameter'.

Time:Monday, February 9, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Richard Balka (UW)

Title:Restrictions of Brownian Motion and Monotone Subsequences of Random Walks

Abstract:It is well known that the level sets and the set of record times of a linear Brownian motion B have Hausdorff dimension 1/2 almost surely. Can we find a larger random set A on which Brownian motion is monotone? Perhaps surprisingly, the answer is negative. I will show that, almost surely, there is no set A such that dim A >1/2 and B: A \to R is monotone. The proof is an application of Kaufman's dimension doubling theorem for planar Brownian motion. More generally, I will discuss how large a set A can be on which Brownian motion is of bounded variation or \alpha-Holder continuous for some \alpha > 1/2. This is a joint work with Yuval peres. Analogous results for fractional Brownian motion, deterministic self-affine functions and random walks will be given. This is joint work with Omer Angel, Andras Mathe and Yuval Peres.

Time:Monday, January 26, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Todd Kemp (UCSD)

Title:Unitary Brownian Motion in High Dimension: Strong Convergence and Fluctuations

Abstract:I will address the current state of the art in the large-N limit behavior of the Brownian motion on the rank N unitary group. The empirical eigenvalue distribution has been known to converge to a(n interesting) limit distribution since the late 1990s. In this talk, I will address two finer question about this convergence: what is the precise rate, and what is the noise profile of the fluctuations at this rate? And what happens at the edge of the spectrum? The first question was answered in part by Levy and Maida in 2010, where they showed that the fluctuations for each fixed time are Gaussian; in recent joint work with Cebron, we have shown this holds true for the whole process. The second question is newly answered in my current joint work with Collins and Dahlqvist, where we show that the spectrum converges strongly (in random matrix parlance, it has a "hard edge").

Time:Monday, January 12, 2015 at 2:30 pm.

Location:LOW 116

Speaker:Matt Lorig (Applied Math., UW)

Title:Asympotics for Kolmogorov backward and Hamilton-Jacobi-Bellman PDEs

Abstract:Let X be a Markov diffusion process and define u(t,x) := E[f(X(T)) | X(t) = x]. Under mild conditions, the function u is the solution of a Kolmogorov backward equation -- a linear parabolic PDE.

Similarly, let Y be a controlled Markov diffusion and define V(t,y) := sup E[ f(Y(T)) | Y(t) = y ], where the sup is taken over all admissible controls. Under certain conditions, the function V is the solution of a Hamilton Jacobi Bellman (HJB) equation -- a (highly) nonlinear PDE.

Closed-form solutions to Komogorov and HJB PDEs are rare. In this talk, we will discuss how to obtain closed-form approximate solutions of these PDEs.

Time:Monday, December 1, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Sebastien Bubeck (Princeton and Microsoft Research)

Title:ON THE INFLUENCE OF THE SEED IN THE PREFERENTIAL ATTACHMENT AND UNIFORM ATTACHMENT MODELS

Abstract:We are interested in the following question: suppose we generate a large tree according to either the preferential attachment model (a.k.a. the plane-oriented random recursive tree), or the uniform attachment model (a.k.a. the uniform random recursive tree)---can we say anything about the initial (seed) tree? A precise answer to this question could lead to new insights for the diverse applications of these models. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we prove that different seeds lead to different distributions of limiting trees from a total variation point of view (part of the proof comes from a recent work of Curien et al.).

Time:Monday, November 24, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Mark Meerschaert (Michigan State University)

Title:TEMPERED FRACTIONAL CALCULUS

Abstract:Fractional derivatives and integrals are (distributional) convolutions with a power law. Including an exponential term leads to tempered fractional derivatives and integrals. Tempered fractional Brownian motion, the tempered fractional integral or derivative of a Brownian motion, is a new stochastic process whose increments can exhibit semi-long range dependence. The tempered finite difference operator is also useful in time series analysis, where it provides a useful new stochastic model for turbulent velocity data. Tempered stable processes are the limits of random walk models, where the power law probability of long jumps is tempered by an exponential factor. These random walks converge to tempered stable stochastic process limits, whose probability densities solve tempered fractional diffusion equations. Tempered power law waiting times lead to tempered fractional time derivatives. Applications include geophysics and finance.

UW-PIMS Mathematics Colloquium

Time:Friday, November 21, 2014 at 2:30 pm.

Location:SIG 225

Speaker:Andrea R. Nahmod (University of Massachusetts, Amherst)

Title:RANDOMIZATION AND LONG TIME DYNAMICS IN NONLINEAR EVOLUTION PDE

Abstract:In the last two decades, there has been substantial progress in the study of nonlinear dispersive PDE thanks to the influx of ideas and tools from nonlinear Fourier and harmonic analysis, geometry and analytic number theory, to the existing functional analytic methods. This body of work has primarily focused on deterministic aspects of wave phenomena and answered important questions related to existence and long time behavior of solutions in various regimes. Yet there remain important obstacles and open questions.

A natural approach to tackle some of them, and one which has recently seen a growing interest, is to consider certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem, invariant measures, etc) and incorporate to the deterministic toolbox, powerful but still classical tools from probability as well. Such approach goes back to seminal work by Bourgain in the mid 90's where global well-posedness of certain periodic Hamiltonian PDEs was studied in the almost sure sense via the existence and invariance of their associated Gibbs measures.

In this talk we will explain these ideas, describe some recent work and future directions. with an emphasis on the interplay of deterministic and probabilistic approaches.

Boeing Lecture

Time:Thursday, November 20, 2014 at 4 pm.

Location:Smith Hall room 205

Speaker:Erhan Bayraktar (University of Michigan)

Title:Stochastic Perron's method for Hamilton-Jacobi-Bellman equations

Abstract:We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using Stochastic Perron's method, we construct a super-solution dominating it. A comparison argument easily closes the proof. The program has the precise meaning of verification for viscosity-solutions, obtaining the DPP as a conclusion. We will also discuss the effectiveness of this method in analyzing the robust feedback-type optimal control problems.

Time:Monday, November 17, 2014 at 2:30 pm.

Location:LOW 117

Speaker:RANDOM WALKS ON METRIC MEASURE SPACES

Title:Mathav Murugan (Cornell University)

Abstract:A metric space is a length space if the distance between two points equals the infimum of the lengths of curves joining them. For a measured length space, we characterize Gaussian estimates for iterated transition kernel for random walks and parabolic Harnack inequality for solutions of a corresponding discrete time version of heat equation by geometric assumptions (Poincare inequality and Volume doubling property). Such a characterization is well known in the setting of diffusion over Riemannian manifolds (or more generally local Dirichlet spaces) and random walks over graphs (due to the works of A. Grigor'yan, L. Saloff-Coste, K. T. Sturm, T. Delmotte, E. Fabes & D. Stroock). However this random walk over a continuous space raises new difficulties. I will explain some of these difficulties and how to overcome them. We will discuss some motivating examples and applications.

Time:Monday, November 10, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Soumik Pal (University of Washington)

Title:SPECTRAL DYNAMICS OF RANDOM REGULAR GRAPHS AND THE POISSON FREE FIELD

Abstract:A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider d many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree 2d. We consider the following dynamics. The 'dimension' of each permutation grows by coupled Chinese Restaurant Processes, while in 'time' each permutation evolves according to the random transposition Markov chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. We give a Poisson random surface description in dimension and time of the limiting cycle counts for every d. As d grows to infinity, the fluctuation of the limiting cycle counts, across dimension, converges to the Gaussian Free Field. When time is run infinitesimally slowly, this field is preserved by a stationary Gaussian dynamics. The laws of these processes are similar to eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes. Part of this talk is based on joint work with Toby Johnson and the rest is based on joint work with Shirshendu Ganguly.

Time:Monday, November 3, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Dan Lanoue (University of California, Berkeley)

Title:A SPATIAL GENERALIZATION OF KINGMAN'S COALESCENT

Abstract:The Metric Coalescent (MC) is a measure-valued Markov Process generalizing the classical Kingman Coalescent. We show how the MC arises naturally from a discrete agent based model (the Compulsive Gambler process) of social dynamics and prove an existence and uniqueness theorem extending the MC to the space of all Borel probability measures on any locally compact Polish space. We'll also look in depth at the case of the MC on the unit interval.

2014 Pacific Northwest Probability Seminar

Time:Saturday, October 25 at Microsoft Research

Conference website:http://research.microsoft.com/nwprob2014/9:00 - 11:15 Coffee and muffins

11:15 - 11:50 talk -- Alexander Holroyd

12:00 - 12:35 talk -- Tim Hulshof

12:40 - 1:40 Lunch (catered)

1:40 - 2:15 Probability demos and open problems

2:20 - 3:15 Birnbaum lecture -- Srinivasa Varadhan

3:25 - 4:00 talk -- Benjamin Young

4:05 - 4:35 Tea and snacks

4:35 - 5:10 talk -- Ioana Dumitriu

6:00 - Dinner (no-host)

Titles

Alexander E Holroyd (Microsoft Research):Finitely dependent coloring

Tim Hulshof (University of British Columbia):Sharp higher order corrections for the critical value of a bootstrap percolation model

Srinivasa Varadhan (New York University) (Birnbaum lecture):The role of compactness in large deviations

Benjamin Young (University of Oregon):A Markov growth process for Macdonald's identity

Ioana Dumitriu (University of Washington):A Regular Stochastic Block Model

Time:Monday, October 20, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Christopher Hoffman (University of Washington)

Title:OIL AND WATER

Abstract:We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting with $n$ ``oil'' and $n$ ``water'' particles at the origin, the particles diffuse in $\bf Z$ according to the following rule: whenever some site $x \in\bf Z$ has at least 1 oil and at least 1 water particle present, it fires by sending 1 oil particle and 1 water particle each to an independent random neighbor $x \pm 1$. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence.

This is joint work with Elisabetta Candellero, Shirshendu Ganguly and Lionel Levine.

Time:Monday, October 13, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Xicheng Zhang (Wuhan University, China)

Title:FUNDAMENTAL SOLUTION OF KINETIC FOKKER-PLANCK OPERATOR WITH ANISOTROPIC NONLOCAL DISSIPATIVITY

Abstract:By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic Levy process, and the drift term is allowed to be cubic growth.

Time:Monday, October 6, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Russell Lyons (Indiana University)

Title:DETERMINANTAL PROBABILITY: SURPRISING RELATIONS

Abstract:(1) For each subset $A$ of the circle with measure $m$, there is a sequence of integers of Beurling-Malliavin density $m$ such that the set of corresponding complex exponentials is complete for $L^2(A)$. (2) Given an infinite graph, simple random walk on each tree in the wired uniform spanning forest is a.s. recurrent. (3) Let $Z$ be the set of zeroes of a random Gaussian power series in the unit disk. Then a.s., the only function in the Bergman space that vanishes on $Z$ is the zero function. (4) In our talk, we explain a theorem that has (1) and (2) as corollaries. We also describe a conjectural extension that has (3) (which is not known) as a corollary. All these depend on determinantal probability measures. All terms above will be explained.

Time:Monday, September 29, 2014 at 2:30 pm.

Location:LOW 117

Speaker:Douglas Rizzolo (Univeristy of Washington)

Title:A GAS PARTICLE IN A GRAVITATIONAL FIELD

Abstract:We will discuss the motion a tagged gas particle in a gravitational field. Our starting point will be a Markov approximation to a Lorentz gas model with variable density. We investigate how the density of the ambient gas impacts the recurrence or transience of the tagged particle. Additionally, we will show that there are multiple scaling regimens leading to nontrivial diffusive limits. This talk is based on joint work with Krzysztof Burdzy.

Time:Monday, June 2, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Adityanand Guntuboyina (UC Berkeley)

Title:Sharp inequalities between f-divergences

Abstract:f-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler divergence, chi-squared divergence, squared Hellinger distance, total variation distance etc. This talk will be about the problem of maximizing or minimizing an f-divergence subject to a finite number of constraints on other f-divergences. We show that these infinite-dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable. Our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f-divergences. This is joint work with Sujayam Saha and Geoffrey Schiebinger.

Joint with

UW-PIMS Mathematics Colloquium

Time:Monday, May 19, 2014 at 2:30 pm.

Location:Loew Hall 106

Speaker:Shige Peng (Shandong University)

Title:Brownian motion under Knightian uncertainty and path-dependent risk

Abstract:A typical measure of risk in finance must take into account of the uncertainty of probability model itself (called Knightian uncertainty). Nonlinear expectation and the corresponding non-linear distributions provides a deep and powerful tool: cumulated nonlinear i.i.d random variables of order 1/n tend to a maximal distribution, according a new law of large number, whereas, the accumulation of order 1/\sqrt{n} tends to a nonlinear normal distribution which is the corresponding central limit theorem. The continuous time uncertainty cumulation forms a nonlinear Brownian motion.

The related stochastic calculus under Knightian uncertainty provides us a powerful tool of valuation for path-dependent derivatives. The corresponding Feynman-Kac formula gives one to one correspondence between fully nonlinear parabolic partial differential equations and backward stochastic differential equations driven by the nonlinear Brownian motion.

Time:Monday, May 12, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Harry Crane (Rutgers University)

Title:Characterization theorems and convergence rates for partition-valued Markov chains

Abstract:We discuss some aspects of exchangeable Markov chains on spaces of set partitions. Specifically, we obtain a Levy-Ito-type characterization of all such Markov chains that possess the Feller property. This characterization elicits a representation in termsof products of random matrices, from which the cutoff phenomenon follows for a large subclass.

Time:Monday, May 5, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Wai-Tong (Louis) Fan (University of Washington)

Title:An interacting particle system at two different scales

Abstract:Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce a class of interacting particle systems that can model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales of observations, we prove the hydrodynamic limits and the fluctuation limits for the systems. Proving these two types of limits represents establishing the law of large numbers and the central limit theorem, respectively, for the time-trajectory of the particle densities. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is joint work with Zhen-Qing Chen.

Time:Monday, April 28, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Hari Narayanan (UW)

Title:Testing the Manifold Hypothesis

Abstract:We are confronted with very high dimensional data sets. As a result, methods of dealing with high dimensional data have become prominent. One geometrically motivated approach for analyzing data is called manifold learning. The underlying hypothesis of this subfield of machine learning is that high dimensional data tend to lie near a low dimensional manifold. However, the basic question of understanding when data lies near a manifold ispoorly understood. I will describe joint work with Charles Fefferman and Sanjoy Mitter ondeveloping a provably correct algorithm to test this hypothesis using i.i.d. samples from an arbitrary distribution supported in the unit ball in a Hilbert space.

Joint Probability and Rainwater seminar

Time:Tuesday, April 22, 2014, 1:30-3:30 pm.

Location:SMI 307

Speaker:Jason Miller (MIT)

Title:Random Surfaces and Quantum Loewner Evolution

Abstract:What is the canonical way to choose a random, discrete, two-dimensional manifold which is homeomorphic to the sphere? One procedure for doing so is to choose uniformly among the set of surfaces which can be generated by gluing together $n$ Euclidean squares along their boundary segments. This is an example of what is called a random planar map and is a model of what is known as pure discrete quantum gravity. The asymptotic behavior of these discrete, random surfaces has been the focus of a large body of literature in both probability and combinatorics. This has culminated with the recent works of Le Gall and Miermont which prove that the $n \to \infty$ distributional limit of these surfaces exists with respect to the Gromov-Hausdorff metric after appropriate rescaling. The limiting random metric space is called the Brownian map.

Another canonical way to choose a random, two-dimensional manifold is what is known as Liouville quantum gravity (LQG). This is a theory of continuum quantum gravity introduced by Polyakov to model the time-space trajectory of a string.Its metric when parameterized by isothermal coordinates is formally describedby $e^{\gamma h} (dx^2 + dy^2)$ where $h$ is an instance of the continuum Gaussian free field, the standard Gaussian with respect to the Dirichlet inner product.Although $h$ is not a function, Duplantier and Sheffield succeeded in constructingLQG rigorously as a random area measure. LQG for $\gamma=\sqrt{8/3}$ is conjecturally equivalent to the Brownian map and to the limits of other discrete theories of quantum gravity for other values of $\gamma$.

In this talk, I will describe a new family of growth processes calledquantum Loewner evolution (QLE) which we propose using to endow LQG with a distancefunction which is isometric to the Brownian map. I will also explain how QLE is related to DLA, the dielectric breakdown model, and SLE.

Based on joint works with Scott Sheffield

Time:Monday, April 14, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Andrey Sarantsev (University of Washington)

Title:INFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES

Abstract:Consider an infinite system of ranked Brownian particles on the real line. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its rank. When two particles collide, they do this is an asymmetric fashion: the local time of collision is split in some fixed proportion between them, depending on their ranks. And this proportion is not necessarily even. We show existence and uniqueness results. We find stationary distributions for the gap process and show convergence results. Our technique is stochastic comparison. This talk is a sequel to the General Exam talk.

Rainwater Seminar

Time:Tuesday, April 8, 2014. Part I: 1:30-2:00, Part II: 2:15-3:15

Location:Padelford C-401

Speaker:Zhen-Qing Chen (University of Washington)

Title:Anomalous diffusions and fractional order differential equations

Abstract:Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. In the first 30 minutes of the talk, I will discuss the connections between anomalous diffusions and differential equations of fractional order. In the second part of the talk, I will present in a gentle way some recent results on the heat kernels for differential operators of fractional order.

Time:Monday, April 7, 2014 at 2:30 pm.

Location:MEB 248

Speaker:Ronen Eldan (Microsoft Research)

Title:Diffusion-limited aggregation on the hyperbolic plane

Abstract:Diffusion-limited aggregation (DLA) is a process in which particles subject to motion by a random walk cluster together to form aggregates. In the Euclidean setting, many fundamental questions about this process such as the rate of growth of its diameter and the density of the final cluster appear to be notoriously hard and have been open for a few decades. We study an analogous version of this process in hyperbolic geometry and show that the aggregate at time infinity has a positive density, unlike what is conjectured for the Euclidean case.

Time:Monday, March 31, 2014 at 2:30 pm.

Location:MEB 238

Speaker:Mike Cranston (U. of California at Irvine)

Title:Self-adjoint extensions, point potentials, and pinned polymers

Abstract:In this talk we discuss closed self adjoint extensions of the Laplacian and fractional Laplacian on L^2 of Euclidean space minus the origin. In many cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. Using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. The talk is based on joint works with Koralov, Molchanov, Squartini and Vainberg.

Time:Monday, March 10, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Matthew Junge (University of Washington)

Title:Frog Model on Trees

Abstract:Put a sleeping frog at each vertex of a graph. At time 0 one of these frogs wakes up and begins a simple random walk in discrete time. Whenever it visits a sleeping frog, the sleeping frog wakes up and begins its own independent walk, also waking any sleepers it visits.

The burning question is: How fast are these frogs waking up? Ill present Toby and my best effort at addressing this question when the underlying graph is a rooted (possibly infinite) d-ary tree. Along the way we partially solve a decade old open problem from Sergui Popov and a question more recently posed by Itai Benjamini.

Joint with Toby Johnson.

Rainwater Seminar

Time:Tuesday, March 4, 2014. Part I: 1:30-2:00, Part II: 2:15-3:15

Location:Padelford C-401

Speaker:Jun Kigami (Kyoto University)

Title:Volume doubling property, quasisymmetry and heat kernel estimates --- Analysis and geometry of self-similar sets

Abstract:In the first part, I will give some fundamental results about analysis on fractals: construction of Brownian motion, heat kernel estimates and distribution of eigenvalues, for example. In the second part, I will focus on time changes of the Brownian motions on Sierpinski carpets. If the speed measure has the volume doubling property, I will show that we can construct a metric which is quasisymmetric to the Euclidean metric and under which we have sub-Gaussian heat kernel estimate.

Time:Monday, March 3, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Soumik Pal (University of Washington)

Title:The geometry of arbitrage: How to gamble (in the stock market) if you must.

Abstract:The theory of gambling goes back to the roots of probability theory. We consider a modern version of it which is model-free and played in the stock market. Suppose we do not model how stock prices will evolve in the future. Is it possible, by active trading, to do better than a market index (say, Dow Jones)? The answer to this question is surprisingly tight. If we restrict ourselves to strategies that are functions of the current stock prices, there is exactly one class that achieves this goal. More remarkably, these strategies are produced as solutions of Monge-Kantorovich optimal transport problem on the multidimensional unit simplex with a cost function that can be described as log partition function. The ideas are an interplay between convex analysis, probability theory, and the geometry of the unit simplex. If time permits we will talk about the mathematics of statistical arbitrage in high-frequency trading. Based on joint work with Leonard Wong.

Time:Monday, February 24, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Hubert LaCoin (CEREMADE in Université Paris Dauphine)

Title:Cutoff for adjacent transpositions and exclusion process on the segment

Abstract:The adjacent transposition shuffle can be described as follows: we have a deck of N card and at each step we select a card at random and exchange its position with the card above it. We wonder how many step are necessary to mix the pack of card with such a shuffle. For the continuous time version of the model we show that around time N^2\log N/(2\pi^2), the total-variation distance to equilibrium of the deck distribution drops abruptly from 1 to 0, and that the separation distance has a similar behavior but with a transition occurring at time (N^2\log N)/\pi^2. This solves a conjecture formulated by David Wilson. The method for the proof uses techniques developed for spin systems including FKG inequality and a censoring inequality due to Peres and Winkler.

Time:Monday, February 10, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Janos Engländer (University of Colorado)

Title:Local vs. global growth for spatial branching processes

Abstract:We will consider spatial branching processes (discrete particle systems and superprocesses) and review some recent results and examples that concern the large time growth (or decay) of these processes -- both locally and globally. Exponential as well as super-exponential growth will be studied. These results are related to spine techniques and also to the spectral and gauge theories of elliptic operators with potentials.

The talk is based on a joint project with Z-Q. Chen (Seattle) and on another one with Y. Ren (Beijing) and R. Song (Urbana).

Time:Monday, February 3, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Wai-Tong (Louis) Fan (University of Washington)

Title:Systems of reflected diffusions with annihilations through membranes

Abstract:We study interacting particle systems which can model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. The hydrodynamic limit and the fluctuation limit for the particle densities can be described, respectively, by a coupled Partial Differential Equation (PDE) and a Gaussian process solving a Stochastic Partial Differential Equation (SPDE). New tools of discrete approximations to reflected diffusions will be discussed.

This is joint work with Zhen-Qing Chen.

Time:Monday, January 27, 2014 at 2:30 p.m.

Location:Johnson 026

Speaker:Erik Slivken (University of Washington)

Title:Pattern avoiding permutations and Brownian excursion

Abstract:Permutations of size n that avoid a given pattern of length 3 can be counted by the nth Catalan number. Dyck paths of length 2n can also be counted by the nth Catalan number. There exist many bijections between the two sets. Remarkably, given the right choice of bijection, both of these random objects converge (in some sense) to the same thing, Brownian excursion. The convergence to Brownian excursion helps answer questions about the pattern avoiding permutation, such as the number of fixed points of permutation.

This is joint work with Christopher Hoffman and Douglas Rizzolo.

Time:Monday, January 13 2:30 p.m.

Location:Johnson 026

Speaker:Panki Kim (Seoul National University)

Title:On Potential theory of Subordinate Brownian motion : Stable and Beyond

Abstract:Subordinate Brownian motion (SBM) is a Levy process which can obtained by replacing the time of Brownian motion by an independent increasing Levy process. In this talk, we introduce a unified method to cover a large class of subordinate Brownian motions including geometric stable process. Recent results on estimates of the Green functions, Poisson Kernels and boundary behavior of harmonic functions with respect to SBM will be discussed.

Time:Monday, December 2, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Eyal Lubetzky (Microsoft Research)

Title:Harmonic pinnacles in the Discrete Gaussian model

Abstract:The 2D Discrete Gaussian model gives each height function $\eta: Z^2 \to Z$ a probability proportional to $\exp[-\beta H(\eta)]$, where $\beta$ is the inverse-temperature and $H (\eta) = \sum (\eta_x-\eta_y)^2$ sums over nearest-neighbor bonds $(x,y)$. We consider the model at large fixed $\beta$, where it is flat unlike its continuous analog (the Gaussian Free Field).

We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ where $M\sim [(2/\pi\beta)\log L\log\log L]^{1/2}$. The key is a large deviation estimate for the height at the origin in $Z^2$, dominated by ``harmonic pinnacles'', integer approximations of a harmonic variational problem. Second, in this model conditioned on $\eta\geq 0$ (a floor), the average height rises, and in fact the height of almost sites concentrates on levels $H,H+1$ where $H\sim M/\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Frohlich (1986). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices.

Joint work with Fabio Martinelli and Allan Sly.

Time:Monday, November 25, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Panki Kim (Seoul National University)

Title:Parabolic Littlewood-Paley inequality for $\phi(-\Delta)$-type operators and applications to Stochastic integro-differential equations

Abstract:In this talk we introduce a parabolic version of the Littlewood-Paley inequality for the operators of the type $\phi(-\Delta)$, where $\phi$ is a Bernstein function.As an application, we construct an $L_p$-theory for the stochastic partial integro-differential equations of the type$du=(-\phi(-\Delta)u+f)dt +gdW_t$. This is a joint work with Ildoo Kim and Kyeonghun Kim.

Time:Monday, November 18, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Andrey Sarantsev (University of Washington)

Title:INFINITE-DIMENSIONAL REFLECTED BROWNIAN MOTION AND COMPETING PARTICLE SYSTEMS

Abstract:Consider the following infinite system of Brownian particles on the real line. The lowest particle moves as a Brownian motion with drift one and variance one, and all other particles move as standard Brownian motions. We are interested in the stationary distributions for the gap process. Pal & Pitman (2008) found one of them: product of exponentials with rate two. We find an infinite family of stationary distributions which includes the one just mentioned. Our main tool is reflected Brownian motion in the infinite-dimensional orthant.

Time:Monday, November 4, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Brent Werness (University of Washington)

Title:Alternate constructions of the Gaussian free field and fast simulation of Schramm-Loewner evolutions

Abstract:The Schramm--Loewner evolutions (SLE) are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models. SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather difficult to produce large, high-fidelity simulations of the process due to the significant correlation between segments of the simulated curve. The standard simulation method works by discretizing the construction of SLE through the Loewner ODE, leading to a O(N^2) algorithm (where N is the length of the SLE curve).

Recent work of Sheffield and Miller has provided an alternate description of SLE, where the curve generated is taken to be a flow line of the vector field $e^{ih}$, where h is a Gaussian free field. In this talk I will describe a new method of approximately sampling a Gaussian free field, and show how this allows us to much more efficiently simulate an SLE curve (O(N\log(N))).

Time:Monday, October 28, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Christopher Hoffman (University of Washington)

Title:The spectrum of Erdos-Renyi random graphs near the connectivity threshold

Abstract:We study the spectrum of the normalized Laplacian of the giant component of an Erdos-Renyi random graphs. Both above and below the connectivity threshold we show that the spectral gap is on the order of 1+O((Average Degree)^{-.5}). We will briefly discuss how these results have applications to random topological spaces.

Time:Monday, October 14, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Tvrtko Tadic (University of Washington)

Title:Stochastic heat equation as a limit of a Brownian motion indexed by a rhombus graph

Abstract:The plane is tiled by cells of specific form that make up a graph, and on the (representation of) edges of this graph we define a process. The question is what happens to the process when the cell size goes to zero? Is there a process in the limit? What will that process (indexed by the whole plane) be?

Burdzy and Pal in their paper studied such a process on a honeycomb graph for a Brownian motion type process, and got that the covariance structure is non-trivial in the limit.

We will talk about a rhombus graph for the same process, and show what happens in two cases when the size of the cell goes to zero. We will also comment how to interpret of the results.

Time:Monday, October 7, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Soumik Pal (University of Washington)

Title:Intertwining diffusions and wave equations

Abstract:An intertwining of two Markov chains is a coupling of the two processes whereby one chain can be sampled from a fixed distribution by running the other. An analogous definition exists for diffusion processes. Although intertwining and the related concept of duality have been around for a long time, recently, there has been an upsurge in interest in new intertwined systems related to models of random matrices and random surfaces. The delicate construction of intertwined chains in this context hinge on skew-Cauchy identities satisfied by symmetric polynomials. For diffusions, their occurrences were a complete mystery and only a few miraculous examples (such as the Brownian-Bessel intertwining, the Warren process and the Whittaker model associated with the Hamiltonian of the quantum Toda lattice) were known in the literature.

We will present a complete characterization of intertwined diffusions through solutions of hyperbolic partial differential equations (e.g., the classical wave equations). This approach covers all the known examples, and leads to a systematic way of generating families of new ones. Moreover, this construction can be thought of as the first probabilistic representation of solutions of second order hyperbolic PDEs.

Joint work with Mykhaylo Shkolnikov.

Time:Monday, September 30, 2013 at 2:30 pm.

Location:MEB 243

Speaker:Krzysztof Burdzy (University of Washington)

Title:On meteors, earthworms and WIMPs

Abstract:I will discuss a model of mass redistribution on graphs. Each vertex initially holds some positive amount of mass. Meteors hit different sites (vertices) according to independent Poisson processes. I will focus on the stationary distribution - its existence, uniqueness and properties. The earthworm model assumes that the mass redistribution events are triggered by an earthworm--a symmetric random walk on the graph. A WIMP is a pair of "weakly interacting mathematical particles". WIMPs are an essential tool in the analysis of the stationary distribution in our model.

Joint work with Sara Billey, Soumik Pal and Bruce Sagan.

Time:Monday, June 3, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Jeffrey Steif (MSR and Chalmers)

Title:Mixing times for random walk on dynamical percolation

Abstract:We study the behavior of random walk on dynamical percolation. In dynamical percolation, the edges of a graph $G$ are either open or closed independently and then evolve in time by "closed changing to open at rate $p\mu$" and "open changing to closed at rate $(1-p)\mu$". We then run a rate 1 random walk in this randomly evolving graph.

If the graph is the discrete torus in $Z^d$ with side length $n$, we study the "mixing time" which is the time it takes for the walker to become approximately uniformly distributed. It turns out that if $p$ is smaller than the critical value for $Z^d$ percolation, then the mixing time is of order $n^2/\mu$ while the time is much shorter when $p$ is larger than the critical value. Other results will also be described.

All definitions, such as percolation and mixing times, will be given. This is joint work with Yuval Peres and Alexandre Stauffer.

Time:Monday, May 20, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Amarjit Budhiraja (UNC)

Title:Long Time Asymptotics of Constrained Diffusions in Polyhedral Cones

Abstract:We consider a family of reflected jump-diffusions in polyhedral cones. Sufficient conditions for transience and positive recurrence are identified. Results on rates of convergence to invariant measures and on their numerical approximations are presented. Asymptotics of exit times from bounded domains, when the diffusion coefficient is scaled by a small parameter, are studied.

Time:Monday, May 13, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Louis Fan (University of Washington)

Title:Hydrodynamic limit of interacting particle systems

Abstract:Hydrodynamic limit provides the link connecting the microscopic behavior and the macroscopic behavior of interacting particle systems. We will provide various examples to illustrate this connection and the general techniques involved, and then discuss two new Reaction-Diffusion type systems which associate to the same system of PDEs coupled through the boundary. (Joint work with Zhen-Qing Chen)

Time:Monday, May 6, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Elliot Paquette (University of Washington)

Title:Hamiltonian cycles in random regular graphs and the small subgraph conditioning method

Abstract:In a line of work that started in the 80s with BollobĂĄs, Fenner, and Frieze and culminated into the 90s, it was shown by Robinson and Wormald that a uniformly chosen d-regular graph on n vertices has a Hamiltonian cycle with probability going to 1 as n goes to infinity. Their method is known as small subgraph conditioning. Informally this can be described as saying that "all the variance between two random regular graph models comes from small cycles," which we will make precise. With Toby Johnson, we show how to make this small subgraph conditioning method quantitative, so that we can estimate the probability of being Hamiltonian as well as allow d to grow to infinity with n. We develop tools that are modifications of Stein's method and that are likely useful in other contexts.

Time:Monday, April 29, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Douglas Rizzolo (UW)

Title:Scaling limits of Markov branching trees

Abstract:We will discuss recent developments in the field of scaling limits of Markov branching trees. Since scaling limits of random trees were introduced by Aldous in the early 90's, focus has largely been on establishing scaling limits for very particular types of trees, such as Galton-Watson trees and consistent Markov growth models. In this talk we will look at the relatively new general theory of scaling limits of Markov branching trees. Plenty of examples will be given to show how the general theory can be used to gain insight into particular models.

Time:Monday, April 22, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Shirshendu Ganguly (UW)

Title:Escape rates for rotor walks

Abstract:Rotor walk introduced by Jim Propp is a deterministic analogue of the simple random walk on a graph. In this talk we will discuss its recurrence and transience properties on naturally occuring graphs like trees and the integer lattices Z^d under certain initial confiurations. Closeness to random walk on the graph will be established in some sense by looking at the proportion of particles escaping among the first n particles doing the rotor walk. This field contains numerous open problems and we will discuss some of them which occur naturally as a follow up to these results. Results discussed in this talk are based on previous work by Angel and Holroyd and recent joint work with Laura Florescu, Lionel Levine and Yuval Peres.

Time:Monday, April 15, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Leonard Wong (UW)

Title:Boundary theory of random walk and analysis on fractals

Abstract:Analysis on fractals is concerned with generalizing classical analysis to fractal sets. Some central notions are Dirichlet form and its associated Markov process. In this talk, after reviewing some well-known results, we discuss an approach based on the boundary theory of random walk. We discuss the identification of a self-similar set K with the Martin boundary of certain random walks on the symbolic space. This allows us to define objects on the boundary in terms of the random walk. In particular, we will discuss its harmonic measure as well as the existence of induced Dirichlet forms. This is joint work with Ka-Sing Lau.

Time:Monday,April 8, 2013 at 2:30 pm.

Location:PAA-A-114

Speaker:Daniel Valesin (UBC)

Title:Multitype Contact Process on Z: Coexistence, Extinction and Interface

Abstract:We consider a two-type contact process on Z in which both types have equal, finite range and supercritical infection rate. We characterize the set of initial configurations for which one of the types almost surely goes extinct. We show in particular that, for some initial configurations, with positive probability neither of the types goes extinct. We then study the process started from the configuration in which all sites to the left of the origin are occupied by type 1 particles and all sites to the right of the origin are occupied by type 2 particles. We show that the process I(t) given by the size of the interface area between the two types is tight, and the process m(t) defined by the position of this interface converges, under diffusive scaling, to Brownian motion.

Time:Monday, March 11, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Xinxing Chen (Shanghai Jiaotong University)

Title:COLLISIONS OF RANDOM WALKS ON SOME GRAPHS

Abstract:In this talk, we shall discuss whether or not two independent random walks will collide infinitely many times on some graphs. We shall show the different types of percolation, wedge and comb, and see that there is no simple monotonicity for the infinite collision property.

Time:Wednesday, March 6, 2013 at 2:30 pm.

Location:LOW 220

Speaker:Jason Swanson (University of Central Florida)

Title:OINT CONVERGENCE ALONG DIFFERENT SUBSEQUENCES OF THE SIGNED CUBIC VARIATION OF FRACTIONAL BROWNIAN MOTION

Abstract:The signed cubic variation of fractional Brownian motion (fBm) is obtained by considering the sum of the cubes of the increments of fBm over uniformly spaced time intervals whose lengths, Δt, tend to zero. It is well-known that when the Hurst parameter of fBm is set to H=1/6, this sum converges in distribution to a Brownian motion that is independent of the original fBm. In this talk, I will discuss recent joint work with Chris Burdzy and David Nualart in which we study the asymptotic correlation between two distinct sums of this type, where the difference between the two sums is in the value of Δt.

Time:Monday, March 4, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Mykhaylo Shkolnikov (University of California, Berkeley)

Title:DIFFUSIVE LIMITS OF SOME INTERACTING PARTICLE SYSTEMS OF EXCLUSION TYPE

Abstract:We will first review two results on convergence of fluctuations in particle systems of exclusion type: a certain two-dimensional growth model introduced by Borodin and Ferrari and an exclusion process with speed change induced by local interactions of particles. We identify the limits as appropriate Brownian particle systems and point out their connections with random matrix theory and stochastic portfolio theory. Motivated by these connections, we will discuss extensions of the results, in which one obtains multidimensional sticky Brownian motion and more general corners processes as diffusive limits. Such results not only shed light on the asymptotics of the discrete particle systems, but also reveal additional structures in their scaling limits. The talk will be based on joint works with Vadim Gorin, Ioannis Karatzas, Soumik Pal and Miklos Racz.

Time:Monday, February 25, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Panki Kim (Seoul National University)

Title:BOUNDARY HARNACK PRINCIPLE AND MARTIN BOUNDARY AT INFINITY FOR SUBORDINATE BROWNIAN MOTIONS

Abstract:Many aspects of potential theory, such as the Green function estimates, boundary Harnack principle and Martin boundary identification, are known for rather wide classes of subordinate Brownian motion in bounded open sets. On the other hand, except for a few particular examples of Levy processes, much less is known in case of unbounded open sets. In this talk I will discuss some potential theoretic problems for subordinate Brownian motion in unbounded open sets.

This talk is based on the following two joint works with Renming Song and Zoran Vondracek:

1. Global uniform boundary Harnack principle with explicit decay rate and its application http://arxiv.org/abs/1212.3092

2. Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motions http://arxiv.org/abs/1212.3094

Time:Wednesday, February 20, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Sayan Banerjee (University of Washington)

Title:THE BROWNIAN CONGA LINE

Abstract:Imagine a long string or molecule whose tip is being wiggled erratically (performing a Gaussian random walk). Though the randomness kicks in through the tip, the whole string `feels the jerk' as randomness propagates down its length, diminishing with distance from the tip. We investigate a discrete analogue of this motion and show that the process is close in some sense to a smooth random curve. We study its properties, including the distribution of critical points, number of loops, the evolution of loops in time and development of singularities.

Time:Monday, February 11, 2013 at 2:30 pm.

Location:MGH 242

Speaker:David M. Mason (University of Delaware)

Title:STRONG APPROXIMATIONS TO A QUANTILE PROCESS BASED ON n INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

Abstract:Jason Swanson (2007) using classical weak convergence theory proved that an appropriately scaled median of n independent Brownian motion converges weakly to a mean zero Gaussian process. More recently Kuelbs and Zinn (2013) have obtained central limit theorems for a quantile process based n independent copies of certain random processes. These include fractional Brownian motions, which are perturbed to be not zero with probability 1 at zero. Their approach is based on an extension of a result of Vervaat (1972) on the weak convergence of inverse processes. We shall define a quantile process of n independent fractional Brownian motions and discuss strong approximations to it by Gaussian processes. Surprisingly, these approximations are in force on sequences on intervals for which weak convergence cannot hold in the limit. They lead to functional laws of the iterated logarithm via Bahadur-Kiefer representations for these quantile processes. This talk is based on joint work with P\'eter Kevei.

Time:Monday, February 4, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Mary Radcliffe (University of Washington)

Title:GIANT COMPONENTS IN RANDOM GRAPHS WITH RANDOM VERTEX SETS

Abstract:Many models for generating random graphs begin with a fixed collection of vertices, and a rule for defining the probability that two vertices are adjacent. Here, we consider how the analysis of a random graph evolves in certain cases where we allow both the vertex set and edge set to be chosen randomly. In particular, we will consider the eigenvalues of such graphs, and the emergence of the giant component in one such model, the Multiplicative Attribute Graph.

Time:Monday, January 28, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Adrien Kassel (Ecole Normale Superieure)

Title:DETERMINANT OF LAPLACIANS AND RANDOM MULTICURVES ON SURFACES

Abstract:I will explain how to construct scaling limits of naturally weighted cycle-rooted spanning forests on graphs embedded on surfaces and approximating them: this yields probability measures on the space ÎŠ of multicurves of the surface independent of the approximating sequence of graphs. One fundamental tool in this construction is the determinant of the bundle Laplacian. The limiting measures possess some interesting properties which hopefully characterize them in the space of probability measures on ÎŠ. Joint work with Rick Kenyon.

Time:Monday, January 14, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Lerna Pehlivan (York University, Toronto)

Title:RANDOM 312 AVOIDING PERMUTATIONS

Abstract:A permutation of {1,2,âŚ,N} is said to avoid 312 pattern if there is no subsequence of three elements of this permutation that appears at the same relative order as 312. Monte Carlo experiments reveal some features of random 312 avoiding permutations. In light of these experiments we determine some probabilities explicitly.

Time:Monday, January 7, 2013 at 2:30 pm.

Location:MGH 242

Speaker:Mauricio Duarte (Universidad de Chile)

Title:THE AZTEC DIAMOND AND KPZ UNIVERSALITY

Abstract:The KPZ universality class has been an attractive field of research in the past years. The theory is still in its developing years, and many fundamental problems are still open, most remarkably finding an adequate definition of a solution to the KPZ equation in the real line. In this talk we will introduce some discretized examples of the KPZ universality and talk about the difficulties of defining a solution to the KPZ equation, focusing on concepts rather than the mathematical technicalities.

Time:Monday, December 3, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Joe Neeman (UC Berkeley)

Title:Robust Gaussian noise stability

Abstract:Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost maximize this probability must be close to half-spaces.

Time:Monday, November 26, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Geoffrey Grimmett (Cambridge University)

Title:Counting self-avoiding walks

Abstract:How small/large can be the connective constant of a regular graph? We give sharp inequalities for transitive graphs, and we explain how to prove strict inequalities as the graph varies.

Joint work with Zhongyang Li.

Time:Monday, November 19, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Krzysztof Bogdan (Wroclaw University of Technology, Poland)

Title:Eigenvalues of the fractional Laplacian with drift

Abstract:We will add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the effect on the principal eigenvalue for the Dirichlet problem.

Time:Wednesday, November 14, 2012 at 3:40 pm.

Location:LOW 111

Speaker:Tvrtko Tadic (University of Washington)

Title:Properties of processes indexed by time-like graphs

Abstract:Burdzy and Pal in their paper defined Markov processes on a index set that was induced by so called time-like graphs (TLG's). They produced a lot of interesting properties regarding dependencies based on the structure of the graph, and they also showed that for Brownian motion on a honeycomb lattice when the mesh size goes to zero has a non-trivial covariance structure in the limit. In this talk we will expand the family of TLG's, we will deal with a general problem of constructing stochastic processes indexed by TLG's. Some properties will hold in general, and others only if the measures we used for the construction are distributions of Markov processes. At the end we will discuss what happens for the Brownian motion on an n by n net, the connection to the stochastic heat equation, and relation to branching Brownian motion.

Time:Monday, November 5, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Brent Werness (University of Washington)

Title:The parafermionic observable in SLE

Abstract:The Schramm-Loewner Evolutions (SLE) are a recently constructed continuous random process designed to describe the scaling limit of a number of discrete random models from statistical physics. SLE has found great success in this role by being rigorously established as the scaling limit of a number of these discrete models. One of the main tools being used to establish the convergence of these scaling limits is the parafermionic observable. While there has been a large amount of work exploring this observable in these discrete models, comparatively little is known about the observable in the continuum limit of SLE. In this talk, I will present recent work showing that one can make sense of the parafermionic observable for SLE and show how the continuum model can help us understand some of the properties of the discrete version.

Time:Monday, October 22, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Soumik Pal (University of Washington)

Title:Brownian particles with asymmetric collision

Abstract:Imagine a collection of identical particles on the line performing random motion while preserving their order. When two particles collide, classical models assume elastic collision. Informally, if these movements are modeled by Brownian motions, the key stochastic feature of elastic collision is the equal division of the `local time' push among the two colliding particles. However an array of useful particle models behave asymmetrically. A well-known example is the Totally Asymmetric Exclusion Process. We consider such discrete models of particle movements on the line with asymmetric collisions. We show that their scaling limits are given by Brownian particles where the local time of collision is distributed unequally. We identify which of these models have a determinantal transition mechanism. An interesting component of the analysis is a generalization of Skorokhod maps in this context. The limiting continuous processes correct several features of the first order models in stochastic portfolio theory to conform more with available data. Based on joint work with Ioannis Karatzas and Mykhaylo Shkolnikov.

Time:Monday, October 15, 2012 at 2:30 pm.

Location:EEB 031

Speaker:Mark M. Meerschaert (Michigan State University, East Lansing)

Title:Fractional Diffusion: Answers and Questions

Abstract:A fractional diffusion equation replaces the first time derivative and/or the second spatial derivatives(s) in the traditional diffusion equation by fractional derivatives. Space-fractional diffusion equations have stochastic solutions involving (operator) stable laws. Time fractional diffusion equations have stochastic solutions that involve the first passage time of a stable subordinator. Fractional derivatives have recently become very popular in applications to science and engineering. For example, they are popular models for the evolution of pollution plumes in ground water and in rivers. This talk will review the current state of research, and describe several open questions.

Time:Monday, October 8, 2012 at 2:30 pm.

Location:LOW 116

Speaker:Zhen-Qing Chen (University of Washington)

Title:Perturbation by Non-local Operators

Abstract:In this talk, we address the existence and uniqueness of fundamental solutions for fractional Laplacian perturbed by a non-local operator of lower order. We present a necessary and sufficient condition for the fundamental solution to be non-negative, in which case it determines a strong Markov process. We further give two-sided sharp estimates on the fundamental solution.

As a special case, we consider the following stochastic differential equations: $dX_t=dY_t + b(X_{t-})dZ_t$, where Y is a symmetric \alpha-stable process, Z is an independent symmetric \beta-stable process (respectively, a finite range \beta-stable process) with 0< \beta < \alpha <2, and b is a bounded continuous function. We show that the above SDE is uniquely solvable in distribution and its weak solutions form a strong Markov process. We derive two-sided estimates on the transition density function of the process.

Joint work with Jieming Wang.

Time:Monday, October 1, 2012 at 2:30 pm.

Location:LOW 116

Speaker:Hariharan Narayanan (University of Washington)

Title:Randomized Interior Point Methods for Sampling and Optimization

Abstract:Interior point methods are algorithms that optimize convex functions over high dimensional convex sets. From one point of view, an interior point method first equips a convex set with a Riemannian metric and then performs a steepest descent to minimize the objective on the resulting Riemannian manifold. We will describe a randomized variant of an interior point method known as ``the affine scaling algorithm" introduced by I.I.Dikin. This variant corresponds to a natural random walk on the same manifold on which affine scaling would perform steepest descent. We discuss applications to sampling and optimization and prove polynomial bounds on the mixing time of the associated Markov Chain. This talk includes work done in collaboration with Ravi Kannan and Alexander Rakhlin.

Time:Monday, May 21, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Nathaniel Blair-Stahn (University of Washington)

Title:A GEOMETRIC PERSPECTIVE ON FIRST-PASSAGE COMPETITION

Abstract:First-passage competition is a stochastic process modeling two species competing for space in the integer lattice ${\bf Z}^d$, where $d$ is at least 2. The main question of interest is whether there is a positive probability that both species survive indefinitely. That is, can both species eventually conquer an infinite region, or does one species end up completely surrounded by the other one almost surely?

This competition model was introduced by H\"{a}ggstr\"{o}m and Pemantle in 1998 as a generalization of first-passage percolation, which models a single species spreading throughout the graph ${\bf Z}^d$. First-passage percolation is described by the shortest path metric on a graph with random edge weights, and thus it is essentially a model of random geometry. Using large deviations estimates for the so-called Shape Theorem for first-passage percolation in ${\bf Z}^d$, it can be shown that on large scales the stochastic first-passage competition process is well-approximated by an analogous deterministic competition process in Euclidean space ${\bf R}^d$, with high probability. By analyzing the geometry of this limiting deterministic process, I describe the behavior of the random process when one species initially occupies the entire exterior of a cone and the other species initially occupies a single interior site. I use this analysis of competition in cones to strengthen a result of H\"{a}ggstr\"{o}m and Pemantle regarding survival of the two species when each starts at a single point.

Time:Monday, May 14, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Laura Florescu (Los Alamos National Lab)

Title:CONNECTIONS BETWEEN THE ABELIAN SANDPILE MODEL AND THE DIMER MODEL

Abstract:Among the typical models studied in statistical mechanics, such as the Ising, Heisenberg, six-vertex, eight-vertex, XXZ spin chain models, the Abelian Sandpile Model stands out as one which is not as closely explored. This talk will provide an introduction to the ASM model, self-organized criticality, sandpiles in nature and science, as well as connections to other models. In particular, the connection between the Abelian sandpile model and the dimer model on grid graphs will be examined. Results concerning symmetric sandpiles will also be presented through the use of spanning tree and perfect matchings techniques, such as the Temperley and the Kenyon-Propp-Wilson bijections. The talk will end in a presentation of open problems in the ASM, as well as possible connections with other typical models in statistical mechanics.

Time:Monday, May 7, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Mauricio Duarte (University of Washington)

Title:SPINNING BROWNIAN MOTION

Abstract:Obliquely reflected Brownian motion (ORBM) in a domain $D$ is a stochastic process that behaves as Brownian motion inside $D$, but as soon as the process hits the boundary it is pushed back inside $D$ is a prescribed direction that changes through the boundary of $D$. Standard constructions of ORBM involve the submartingale problem and/or the Skorohod problem. We (re)construct ORBM from a non-symmetric Dirichlet form, by using the associated stationary distribution as reference measure of the Dirichlet space.

In the second part of the talk, we present a new reflection process in a bounded, smooth domain $D$ that behaves very much like oblique reflected Brownian motion, except that the directions of reflection depend on an external parameter $S$ called spin. The pair $(X,S)$ is called spinning Brownian motion and is found as the unique strong solution to the following stochastic differential equation:

$$ dX_t = dB_t + \vec\gamma (X_t,S_t)dL_t \qquad (*) $$ $$ dS_t = \left(\vec{g}(X_t) - S_t \right)dL_t \qquad (*) $$ where $\vec\gamma$ points uniformly into $D$, and $L$ is a local time for $X$.

We prove that the solution to (*) has a unique stationary distribution. The main tool of the proof is excursion theory, and an identification of the local time $L$ as a component of an exist system for $X$. I will provide examples to illustrate the proofs of our results.

Time:Monday, April 30, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Joel Barnes (University of Washington)

Title:HYDRODYNAMIC LIMIT OF A BOUNDARY-DRIVEN ELASTIC EXCLUSION PROCESS AND A STEFAN PROBLEM

Abstract:Burdzy, Pal, and Swanson considered solid spheres of small radius moving in the unit interval, reflecting elastically from each other and at $x=0$, and killed at $x=1$, with mass being added to the system from the left at constant rate $a$. By transforming to a system with zero-width particles moving as independent Brownian motion, they derived a limiting stationary distribution for a particular initial distribution, as the width of a particle decreases to zero and the number of particles increases to infinity. This space-removing transformation has a direct analogy in the isomorphism between an unbounded-range exclusion process and a superimposition of random walks with random boundary. We derive the hydrodynamic limits for these isomorphic processes, suggesting that this "elastic" exclusion is an appropriate model for the reflecting Brownian spheres in one dimension.

Time:Monday, April 23, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Janko Gravner (University of California, Davis)

Title:EVOLUTION FROM SEEDS IN ONE-DIMENSIONAL CELLULAR AUTOMATA

Abstract:The talk will give an overview of recent results on simple one-dimensional rules started from seeds, i.e., from bounded perturbations of the quiescent state. Two phenomena, replication and robust periodic solutions emanating from one of the edges, are of particular interest. The talk will emphasize examples and interesting open problems, and will be accessible to undergraduates. (Joint work with D. Griffeath, G. Gliner, and M. Pelfrey.)

Time:Monday, April 16, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Jinqiao Duan (Institute for Pure and Applied Mathematics (IPAM) and Illinois Institute of Technology)

Title:EFFECTIVE DYNAMICS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract:The need to take stochastic effects into account for modeling complex systems has now become widely recognized. Stochastic partial differential equations arise naturally as mathematical models for multiscale systems under random influences. We consider macroscopic dynamics of microscopic systems described by stochastic partial differential equations. The microscopic systems are characterized by small scale heterogeneities (spatial domain with small holes or oscillating coefficients), or fast scale boundary impact (random dynamic boundary condition), among others. Effective macroscopic model for such stochastic microscopic systems are derived. The effective model s are still stochastic partial differential equations, but defined on a unified spatial domain and the random impact is represented by extra components in the effective models. The solutions of the microscopic models are shown to converge to those of the effective macroscopic models in probability distribution, as the size of holes or the scale separation parameter diminishes to zero. Moreover, the long time effectiveness of the macroscopic system in the sense of convergence in probability distribution, and in the sense of convergence in energy are also proved.

Time:Monday, April 9, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Nathanael Berestycki (Cambridge University)

Title:EFFECT OF SELECTION ON THE GENEALOGY OF POPULATIONS

Abstract:We consider random systems of particles which branch and move independently of one another, but are also subject to a selection mechanism that maintains the size of the population essentially constant. Models of this type were recently introduced by physicists Brunet, Derrida and collaborators. Using nonrigorous arguments they derived striking predictions for such systems: notably, the genealogy of the population is given by a universal object, the Bolthausen-Sznitman coalescent. I will give an overview of some of these conjectures and some rigorous recent results in these directions. (Joint work with J. Berestycki and J. Schweinsberg, on the one hand, and L. Zhuo Zhao on the other).

Time:Monday, April 2, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Gregory Miermont (Universite Paris-Sud, Orsay)

Title:THE SCALING LIMIT OF THE MINIMAL SPANNING TREE ON THE COMPLETE GRAPH

Abstract:Assign an independent uniform weight to every edge of the complete graph with $n$ vertices, and let $T_n$ be the minimal spanning tree, i.e. the one which minimizes the sum of weights of the edges it covers. We show that the metric space $n^{-1/3}T_n$, in which the edges of $T_n$ should be thought of as segments of length $n^{-1/3}$, converges in distribution as $n\to\infty$ to a random real tree. The latter seems to be a new model of binary continuum random tree. In particular, its law is singular to that of the Brownian continuum random tree. This talk is based on ongoing joint work with L. Addario-Berry, N. Broutin and C. Goldschmidt.

Time:Monday, March 26, 2012 at 2:30 pm.

Location:SMI 211

Speaker:Toby Johnson (University of Washington)

Title:GROWING RANDOM REGULAR GRAPHS AND THE GAUSSIAN FREE FIELD

Abstract:The spectral properties of Wigner matrices (random symmetric matrices with iid entries above the diagonal) have been studied intensely. The adjacency matrices of random regular graphs have much in common with Wigner matrices, but they can be different too. For example, the fluctuations of their linear eigenvalue statistics converge to sums of Poissons as the size of the graph tends to infinity, rather than to Gaussians as with Wigner matrices.

Alexei Borodin has recently found connections between the eigenvalues of sequences of minors of a Wigner matrix and the Gaussian Free Field. As an analogue to this, we investigate the eigenvalues of a sequence of growing random regular graphs, and we find similar connections. Along the way, we will paint a nice picture of the combinatorial behavior of our growing random regular graphs.

This is joint work with Soumik Pal.

Time:Monday, March 5, 2012 at 2:30 pm.

Location:MEB 235

Speaker:Jason Swanson (UCF)

Title:The calculus of differentials for the weak Stratonovich integral

Abstract:The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of $f(B)$ with respect to $g(B)$, where $B$ is a fractional Brownian motion with Hurst parameter 1/6, and $f$ and $g$ are smooth functions. We use this expression to derive an It\^o-type formula for this integral. As in the case where $g$ is the identity, the It\^o-type formula has a correction term which is a classical It\^o integral, and which is related to the so-called signed cubic variation of $g(B)$. Finally, we derive a surprising formula for calculating with differentials. We show that if $\mathbf{d} M=X\,\mathbf{d} N$, then $Z\,\mathbf{d} M$ can be written as $ZX\,\mathbf{d} N$ minus a stochastic correction term which is again related to the signed cubic variation.

Time:Monday, February 13, 2012 at 2:30 pm.

Location:MEB 235

Speaker:David Mason (Delaware)

Title:Feller Classes and the Asymptotic Behavior of Self-Normalized Sums and Processes

Abstract:Click Here.

Time:Monday, February 6, 2012 at 2:30 pm.

Location:MEB 235

Speaker:James Pfeiffer (UW)

Title:Bootstrap Percolation on the Hamming Torus

Abstract:Bootstrap percolation is a deterministic growth process with a random starting configuration. We consider bootstrap percolation on the Hamming torus and calculate the threshold functions for subgraphs of different dimensions to become open. We show that the threshold function for 1-dimensional subgraphs is distinct from higher dimensional subgraphs, while the threshold functions for $i$ dimensional subgraphs, $i \geq 2$, are very closely bunched.

This is joint work with Chris Hoffman

Time:Monday, Janunary 23, 2012 at 2:30 pm.

Location:MEB 235

Speaker:Dale Roberts (Australian National University)

Title:Equations with Boundary Noise

Abstract:After briefly introducing the concept of stochastic partial differential equations, we concentrate on the special situation where randomness is located on the boundary of a manifold. We suggest, by surveying the existing literature, that this is a particularly natural situation especially in higher dimensions. However, it gives rise to a number of mathematical challenges. In particular, the problem identified in 1993 by Da Prato and Zabczyk: obtaining function-valued solutions in the case of Dirichlet boundary conditions. We conclude by presenting some of our recent results in this direction.

Time:Monday, Janunary 9, 2012 at 2:30 pm.

Location:MEB 235

Speaker:Terry Soo (UVic)

Title:It's Deterministic Poisson Thinning

Abstract:Given a homogeneous Poisson point process it is well known that selecting each point independently with some fixed probability gives a homogeneous Poisson process of lower intensity. This is often referred to as thinning. In this talk we will discuss the following question. Can thinning be achieved without additional randomization; that is, as a deterministic function of the point process, can we choose a subset of the points so that the chosen points from a Poisson process of lower intensity?

On the infinite line and plane (and in any other higher dimension) we can colour a Poisson point process red and blue, so that each colour class forms a Poisson point process; furthermore, the function can be chosen as an isometry-equivariant finitary factor (that is, if an isometry is applied to the points of the original process, the points are still coloured the same way). Thus using only local information, without a central authority or additional randomization, points of a Poisson process can split into two groups, each of which are still Poisson.

On a set of finite volume (say the unit disc or circle), we find that even without considerations of equivariance, thinning can not always be achieved as a deterministic function of the Poisson process and the existence of such a function depends on the intensities of the original and resulting Poisson processes. We prove a necessary and sufficient condition on the two intensities for the existence of such a function. The condition exhibits a surprising lack of monotonicity.

(Joint work with Omer Angel, Alexander Holroyd, and Russell Lyons)

Time:Monday, December 5, 2011 at 2:30 pm.

Location:LOW 102

Speaker:David Wilson (Microsoft Research)

Title:Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on Z^2

Abstract:We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in Z^2, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a 15-year old conjecture about the stationary sandpile density on Z^2. We do the analogous computation for the triangular lattice, honeycomb lattice and Z x R, for which the probabilities are 5/18, 13/36, and 1/4-1/\pi^2 respectively. (Joint work with Richard Kenyon.)

Time:Monday, November 28, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Mounir Zili (Prepratory Institute for Military Academies, Tunisia)

Title:On the Mixed and Sub-Mixed Fractional Brownian Motions

Abstract:We will present the main stochastic properties of the mixed fractional Brownian motion. Then, we will introduce a new Gaussian process that we have chosen to call the "sub-mixed fractional Brownian motion". We will show that this new process can be considered as an intermediate between an ordinary Brownian motion and a mixed fractional Brownian motion.

Time:Monday, November 21, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Daisuke Shiraishi (Kyoto University, Japan)

Title:Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions

Abstract:Let $S^1,S^2$ be independent simple random walks in $Z^d$ (d=2,3) started at the origin. We construct two-sided random walk paths conditioned that $S^1 [0,\infty ) \cap S^2 [1, \infty ) = \emptyset $ by showing the existence of the following limit: $$ \lim _{n \rightarrow \infty } P ( \cdot | S^1 [0, \tau ^1 ( n)] \cap S^2 [1, \tau^2 (n) ] = \emptyset ) =: P^{\sharp}(\cdot ), $$ where $\tau^i (n) = \inf \{ k \ge 0 : |S^i (k) | \ge n \}$. Let $\overline{S}^1, \overline{S}^2$ be the associated two-sided random walks whose probability law is $P^{\sharp}$. We consider the trace $\overline{{\cal G}}=\overline{S}^1 [0,\infty) \cup \overline{S}^2 [0,\infty)$ to be a random subgraph of $Z^d$ (d=2,3) and show that the simple random walk on $\overline{{\cal G}}$ is subdiffusive.

Time:Monday, November 14, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Shuwen Lou (University of Washington)

Title:Multi-dimensional Brownian Motion with Darning

Abstract:The reason that we define multi-dimensional Brownian motion as a darning process is that, even for the simplest case which is R^2 being unioned with R^1, such a process cannot be defined in the usual sense, because 2-dimensional Brownian motion never hits a singleton. Constructions of darning processes are based on one-point extension theory which was first studied by M. Fukushima. Lots of very interesting examples, for instance, circular Brownian motion, Brownian motion with a ``knot", etc., can be constructed in this way, some of which will be provided in the talk. The rest of the talk will be focusing on the heat kernel estimates of multi-dimensional Brownian motion with darning.

Time:Monday, November 7, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Alexandre Stauffer (Microsoft Research)

Title:Space-time percolation and detection by mobile nodes

Abstract:Consider a Poisson point process of intensity \lambda in R^d. We denote the points as nodes and let each node move as an independent Brownian motion. Consider a target particle that is initially placed at the origin at time 0 and can move according to any continuous function. We say that the target is detected at time t if there exists at least one node within distance 1 of the target at time t. We show that if \lambda is sufficiently large, then the target will eventually be detected even if its motion can depend on the past, present and future positions of the nodes. In the proof we use coupling and multi-scale analysis to see this process as fractal percolation and show that some good events percolate in space and time.

Joint Probability and Combinatorics Seminar

Time:Wednesday, November 2, 2011 at 4 pm.

Location:SMI 311

Speaker:Peter Winkler (Dartmouth College and Microsoft Research)

Title:Edge-Cover by Random Walk

Abstract:We show that the time for a random walk to cover all the edges of a graph with m edges is bounded by 2m^2; if all edges must be covered in both directions, 3m^2. These results generalize to graphs with edge-lengths (even with infinitely many vertices) and to Brownian motion. Joint work with Agelos Georgakopoulos.

Time:Monday, October 17, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Russell Lyons (Indiana University)

Title:Metric Spaces of Negative Type in Statistics

Abstract:A classic puzzle asks one to show that given 10 red points and 10 blue points in space, the sum of the 200 distances between oppositely colored points is at least the sum of the 200 distances between like-colored points. Szekely, Rizzo and Bakirov have used this property to produce new algorithms to cluster points, to test multivariate normality, and to test independence. We review their work with particular attention to distance covariance and Brownian covariance, showing how embeddings into Hilbert space clarify matters.There are several probability related talks and event this week (Oct.10 to Oct. 16, 2011).

Special Statistics Seminar

Time: Wednesday Oct. 12, 2011 at 4 pm in Smith Hall 120

Speaker: Persi Diaconis (Stanford University)

Title: Statistical problems in large networks

2011-2012 Common Book Lecture Series

Time: Wednesday, October 12, 2011 at 7:00 PM in Kane Hall 130

Speaker: Persi Diaconis (Stanford University)

Title: The Search for Randomness

UW-PIMS Mathematics Colloquium

Time: Friday Oct. 14, 2011, 2:30 at 2:30pm in MEB 238

Speaker: Joel Spencer (New York University)

Title: Two Needles in Exponential Haystacks

- Pacific Northwest Probability Seminar

Time: Saturday Oct. 15, 2011 from 10:40 am to 5:10 pm at Savery Hall 260

featuring five talks by Steven Evans (Birnbaum Lecture, UC Berkeley), Ori Gurel-Gurevich (UBC), Jason Miller (Microsoft Research), Steffen Rohde (University of Washington), Bartek Siudeja (University of Oregon).

Time:Monday, October 3, 2011 at 2:30 pm.

Location:LOW 102

Speaker:Elliot Paquette (University of Washington)

Title:The Eigenvalues that Fluctuate and the Eigenvalues that Escape Us

Abstract:One of the important problems of modern probability theory is to determine the spectrum of a random matrix. This self-contained talk will introduce some of the central questions of random matrix theory, and show how they are related to one another. Two types of random matrices with very different characteristics will be explored: the $\beta$-ensembles, which might be considered a reinterpretation ofclassical potential theory and admit an analytic approach, and the matrices of random graphs, which require a combinatorial approach. In spite of their differences, both ensembles of matrices show some surprising similarities, which will be presented alongside some new results and open problems.

Time:Monday, September 26, 2011 at 2:30 pm.

Location:PDL C-401

Speaker:Masatoshi Fumushima (Osaka University, Japan)

Title:On general boundary conditions for one-dimensional diffusions in $C_b$ and $L^2$ settings

Abstract:The minimal diffusion $X^0$ on a one-dimensional open interval $I$ is shown to be symmetric with respect to the attached canonical measure $m.$ The Dirichlet form of $X^0$ on $L^2(I;m)$ is identified in terms of the attached triplet $(s, m, k),$ which readily yields descriptions of the $L^2$-generators of $X^0$ and its reflecting extension $X^r.$ By using the associated reproducing kernels, their $C_b$-generators are then indentified and compared with those of Feller, It\^o and McKean.

Time:Monday, May 23, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Allan Sly (Microsoft Research)

Title:Scaling limits of Random Walks on Long Range Percolation Clusters

Abstract:We study limit laws for simple random walks on supercritical long range percolation clusters on Z^d. For the long range percolation model, the probability that two vertices x, y are connected is asymptotically |x-y|^{-s}. For d < s < d+1 we prove that the scaling limit of simple random walk on the infinite component is \alpha-stable Levy motion with alpha = s-d establishing a conjecture of Berger and Biskup.

Time:Monday, May 16, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Masatoshi Fukushima (Osaka University, Japan)

Title:On Brownian motion with darning and Komatu-Leowner equation for multiply connected domains

Abstract:Brownian motion with darning (BMD in abbreviation) is applied to the study of conformal mappings of multiply connected planar domains and the associated Komatu-Leowner differential equations. The notion of BMD and its basic properties are presented in a forthcoming joint book with Z.-Q. Chen in a more general context. It is a diffusion process obtained from the Brownian motion by rendering each hole of the domain into one point. Especially the zero period property of a general BMD-harmonic function and the invariance of BMD under a conformal map up to a time change will be used to derive the continuity of some fundamental quantities related to the Komatu-Leowner equation.This talk is based on an ongoing joint work with Z.-Q. Chen and S. Rohde.

Time:Tuesday, May 10, 2011 at 1:30 pm.

Location:PDL C-401

Speaker:Hamed Amini (Ecole Normale Superieure, France)

Title:Epidemics and percolation in random networks

Abstract:In the first part, we consider bootstrap percolation and diffusion in random graphs, and show how large cascades can be triggered by small initial shocks. We then analyze the impact of the edge weights on distances in sparse random graphs. Our main result consists of a precise asymptotic expression for the weighted diameter when the edge weights are i.i.d. exponential random variables. (Joint work with Marc Lelarge.)

Time:Monday, May 9, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Toby Johnson (University of Washington)

Title:Bipartite Graphs and the Marcenko-Pastur Law

Abstract:The most common distribution in random matrix theory is the semicircle law, which among other things is the limiting distribution of the eigenvalues of a random regular graph, if the number of vertices and degree of each vertex go to infinity. We consider a random biregular bipartite graph, where each class of vertices has a common degree. To find the limiting distribution of the eigenvalues, we need to abandon the semicircle law and turn to the second most common distribution in random matrix theory, the Marcenko-Pastur law. The talk will give some background on random matrix theory before presenting these new results.This is joint work with Ioana Dumitriu.

Time:Monday, May 2, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Janos Englander (University of Colorado at Boulder)

Title:Some particle models with self interaction and in random environment

Abstract:Recently a number of particle models have been studied where individuals move in space and also interact via the center of the system (given by the center of mass). I will review some of my results as well as those of Gill, Balazs and Racz. Time permitting, I will also report on some (simulation) results concerning a branching random walk in a random "cookie" environment. This latter work is joint with N. Sieben.

Joint PDE and Probability Seminar

Time:Thursday, April 28 2011, 10:30 am.--12:30 pm.

Location:PDL C-36

Speaker:Hiroaki Aikawa (Hokkaido University, Japan)

Title:Potential analysis on nonsmooth domains

Abstract:Martin proved that every domain with Green function has the Martin boundary which represents all positive harmonic functions in the domain. The identification of the Martin boundary is a very interesting question and attracts a lot of attentions of numbers of mathematicians.

This talk is devoted to the Martin boundaries of nonsmooth Euclidean domains and related subjects in potential theory. Various interior and exterior conditions on domains are presented with backgrounds. Observe that the boundary Harnack principle and the Carleson estimate play important roles for the identification of the Martin boundary. Technical details such as the box argument are also given.

Time:Monday, April 25, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Subhro Ghosh (UC Berkeley)

Title:What does a Point Process Outside a Domain tell us about What's Inside?

Abstract:In a Poisson point process we have independence between disjoint spatial domains, so the locations of the points outside a disk give us no information on the points inside. The story gets a lot more interesting for processes with stronger spatial correlation. In the case of Ginibre ensemble, a process arising from eigenvalues of random matrices, we prove that the outside points determine exactly the number of points inside, and further, we demonstrate that they determine nothing more. In the case of zero ensembles of a Gaussian power series, we prove that the outside points determine exactly the number and the center of mass of the inside points, and nothing further. These phenomena suggest a certain hierarchy of point processes, where a higher order process is rigid with respect to a greater variety perturbations. Poisson, Ginibre and the Gaussian power series fit in at levels 0, 1 and 2 in this hierarchy.

Time:Monday, April 18, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Hong Qian (University of Washington)

Title:Thermodynamics and Landscape Theory of Nonlinear Stochastic Dynamics and Their Applications to Physics and Biology

Abstract:I shall discuss two recently emerged themes in stochastic modeling of natural phenomena: (1) A thermodynamic superstructure over a Markov process, and (2) A landscape approach to nonlinear stochastic dynamics.

We present results on (1) using finite discrete-state, continuous-time Markov chain by first introducing three concepts: Energy, Entropy and Free Energy. Mathematical relationship among these quantities and their time evolutions constitute a general thermodynamic superstructure of a wide class of Markov processes, possibly generalizable to diffusion processes.

To outline the general theory for (2), we consider continuous-time Markov chain with discrete states defined on Z^n. This model is motivated by biochemical reaction systems in a small volume. In terms of the large deviation rate function for infinite large volume, which we call a "landscape", the law of large numbers, central limit theorem, and multiple time scale phenomenon in nonlinear dynamics will be discussed. We shall also discuss nonlinear dynamics with a limit cycle.

The mathematics will not be rigorous but heuristic via explicit computations. The emphases will be on the possibility for new mathematical research.

Time:Monday, April 11, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Soumik Pal (University of Washington)

Title:The Aldous diffusion on continuum trees

Abstract:Consider a Markov chain on the space of rooted binary trees that randomly removes leaves and reinserts them on a random edge. This chain was introduced by David Aldous in '99 who conjectured a diffusion limit of thischain, as the size of the tree grows, on the space of continuum trees. We talk about how to prove this conjecture. Our approach involves taking an explicit scaled limit which is novel in the area of Markov processes on real trees.

Time:Wednesday, March 30, 2011 at 2:30 pm.

Location:CLK 120

Speaker:Mykhaylo Shkolnikov (Stanford University)

Title:Large systems of interacting diffusion processes

Abstract:We will consider two systems of interacting diffusion processes, which go by the names rank-based and volatility-stabilized models in the mathematical finance literature. We will show that, if one lets the numberof diffusion processes tend to infinity, the limiting dynamics of the system is described by the porous medium equation with convection in the rank-based case and by a degenerate linear parabolic equation in the volatility-stabilized case. In the first case we also provide the corresponding large deviations principle. The results can be applied in stochastic portfolio theory and for the numerical solution of certain partial differential equations. A part of the talk is joint work with Amir Dembo and Ofer Zeitouni.

Time:Monday March 28, 2011 at 2:30 pm.

Location:RAI 116

Speaker:Takashi Kumagai (Kyoto University, Japan)

Title:Convergence of centered Markov chains to non-symmetric diffusions with bounded coefficients

Abstract:We consider a certain class of non-symmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to non-symmetric diffusions. As an application, we approximate non-symmetric divergence forms with bounded coefficients by non-symmetric Markov chains. This extends the results by Stroock-Zheng to the non-symmetric divergence forms.

This is a on-going joint work with J-D. Deuschel (TU-Berlin).

Time:Wednesday, March 9, 2011 at 2:30 pm.

Location:DEN 311

Speaker:Kaneharu Tsuchida (National Defense Academy, Japan)

Title:Large deviation for additive functionals of nearly stable processes

Abstract:We prove the large deviation principle for additive functionals of a Levy process. The Levy process is obtained by a subordinate Brownian motion and the Levy exponent is a nearly stable exponent (stable times slowly varying functions). We prove the large deviation by showing the differentiability of the logarithmic moment generating function and using the Gartner-Ellis theorem.

Time:Monday, March 7, 2011 at 2:30 pm.

Location:LOW 114

Speaker:James Gill (UW)

Title:Random Riemann Surface Classification

Abstract:We discuss the Riemann surface generated by two random surface models and prove a theorem which says that a rather broad class of infinite random triangulations (which includes these models) have recurrent Brownian motion. This is joint work with Steffen Rohde.

Time:Friday, March 4, 2011 at 2:30 pm.

Location:LOW 111

Speaker:Siva Athreya (ISI Bangalore)

Title:Brownian Motion on R trees

Abstract:The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on a given locally compact R-tree equipped with a Radon measure. We then characterize recurrence versus transience. This is joint work with Anita Winter and Michael Eckhoff.

Time:Monday February 28, 2011 at 2:30 pm

Location: LOW 114

Speaker:Mike Hochman (The Hebrew University, Microsoft Research)

Title:Non-polygonal limit shapes in first passage percolation.

Abstract:First passage percolation deals with the large-scale geometry of the randomly weighted graph obtained by placing i.i.d. non-negative weights on the edges of the standard nearest-neighbor graph on the square lattice. It is a classical result by Kesten and others that, under very mild conditions on the marginal weight distribution, if B(r) is the ball of radius r about the origin in the weighted graph metric, then with probability one, B(r)/r converges uniformly to a deterministic compact convex set C. It is conjectured that C is strictly convex in all but the most degenerate cases, and in particular should be non-polygonal. Up until now there have been no examples of weight distributions for which either of these assertions can be verified. I will describe a construction of such a distribution where we can show that the limit shape is not a polygon, and can force any flat edges, if they exist, to be arbitrarily short. Joint work with Michael Damron.

Time:Monday February 14, 2011 at 2:30 pm.

Location: LOW 114

Speaker:Krzysztof Burdzy (UW)

Title:Shy couplings, lion and man, and rubber domains

Abstract:The talk is an exciting mixture of probability, metric geometry, differential games and algebraic geometry. (OK, "rubber band domains" are not quite a topic in algebraic geometry but I thought that "algebraic geometry" would sound sophisticated.) Shy couplings are pairs of processes that do not come close to each other (ever!). Every well educated person should know whether the lion can catch the man. The most surprising claims are: (i) Pursuit problems for Brownian particles (which have "infinite" velocity) are related to pursuit problems with bounded velocities; and (ii) It appears that nobody has thought about "rubber band domains" so far, although they are a very natural concept.

Time:Monday February 7, 2011 at 2:30 pm.

Location: LOW 114

Speaker:Dimitris Achlioptas (UC Santa Cruz)

Title:Algorithmic Barriers from Phase Transitions

Abstract:For many random optimization problems we have by now very sharp estimates of the satisfiable regime. At the same time, though, all known polynomial-time algorithms only find solutions in a very small fraction of that regime. We study this phenomenon by examining how the statistics of the geometry of the set of solutions evolve as constraints are added. We prove in a precise mathematical sense that, for each problem studied, the barrier faced by algorithms corresponds to a phase transition in that problems solution-space geometry. Roughly speaking, at some problem-specific critical density, the set of solutions shatters and goes from being a single giant ball to exponentially many, well-separated, tiny pieces. All known polynomial-time algorithms work in the ball regime, but stop as soon as the shattering occurs. Besides giving a geometric view of the solution space of random optimization problems our results establish rigorously a substantial part of the 1-step Replica Symmetry Breaking picture of statistical physics for these problems.

Time:Monday January 31, 2011 at 2:30 pm.

Location: LOW 114

Speaker:Alexander E Holroyd (Microsoft Research)

Title:Multi-dimensional Percolation.

Abstract:Percolation is concerned with the existence of an infinite path in a (Bernoulli) random subgraph of the lattice Z^D. We can rephrase this as the existence of a Lipschitz embedding (or an injective graph homomorphism) of the infinite line Z into the random subgraph. What happens if we replace the line Z with another lattice Z^d? I'll answer this for all values of the two dimensions d and D, and the Lipschitz constant. The Borsuk-Ulam theorem will make a cameo appearance. Based on joint works with Dirr, Dondl, Grimmett and Scheutzow.

Time:Monday January 24, 2011 at 2:30 pm.

Location: LOW 114

Speaker:Zhen-Qing Chen (UW)

Title:Heat kernel estimates for Dirichlet fractional Laplacian under Gradient Perturbation

Abstract:A rotationally symmetric stable process in Euclidean space with a drift is a strong Markov process X whose infinitesimal generator L is a fractional Laplacian perturbed by a gradient operator. In this talk, I will present recent results on the sharp estimates on the transition density p_{D}(t, x, y) of the sub-process of X killed open leaving a bounded open set D. This transition density function p_{D}(t, x, y) is also the fundamental solution (or heat kernel) of the non-local operator L on D with zero exterior condition. Boundary Harnack principle for X (or, equivalently, L) in C^{2}open sets with explicit boundary decay rate will also be given.

Joint work with P. Kim and R. Song.

Time:Monday January 10, 2011 at 2:30 pm.

Location: LOW 114

Speaker:Elliot Paquette's (UW)

Title:Global Fluctuations for â-Jacobi Ensembles

Abstract:We will present a method to compute global fluctuations for â-Jacobi matrix ensembles. This provides a glimpse into the highly non-local behavior of the eigenvalues of a random matrix, where it is seen that the eigenvalues are very regularly spaced. Starting from a central limit theorem-type result for the traces of powers of matrices, we develop a central limit theorem for a class of continuously differentiable functions applied to the matrices. Along the way, we use Jack functions and combinatorial methods together with results of Anderson-Guionnet-Zeitouni. This is joint work with Ioana Dumitriu.

Time:Monday December 6, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Perla Sousi (Cambridge University)

Title:MOBILE GEOMETRIC GRAPHS: DETECTION, COVERAGE AND PERCOLATION

Abstract:We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in $R^d$ with constant intensity and let each node move independently according to Brownian motion. At any time $t$, we put an edge between every pair of nodes if their distance is at most $r$. We study two features in this model: detection (the time until a target point--fixed or moving--is within distance $r$ from some node of the graph), coverage (the time until all points inside a finite box are detected by the graph) and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain asymptotics for these features by combining ideas from stochastic geometry, coupling and multi-scale analysis. This is joint work with Yuval Peres, Alistair Sinclair and Alexandre Stauffer.

Time:Monday November 29, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Jason Miller (Microsoft Research)

Title:CLE(4) AND THE GAUSSIAN FREE FIELD

Abstract:The discrete Gaussian free field (DGFF) is the Gaussian measure on real-valued functions $h(\,\cdot\,)$ on a bounded subset $D$ of the two dimensional integer lattice, whose covariance is given by the Green's function for simple random walk. The graph of $h(\,\cdot\,)$ is a random surface which serves as a physical model for an effective interface. We show that the collection of random loops given by the level sets of the DGFF at any height converges in the fine-mesh scaling limit to a family of loops which is invariant under conformal transformations when $D$ is a lattice approximation of a non-trivial simply connected domain. In particular, there exists $\lambda>0$ such that the level sets whose height is an odd integer multiple of lambda converges to a nested conformal loop ensemble with parameter $\kappa=4$ [so-called CLE(4)], a conformally invariant measure on loops which locally look like SLE(4). Using this result, we give a coupling of the continuum Gaussian free field (GFF), the fine-mesh scaling limit of the DGFF, and CLE(4) such that the GFF can be realized as a functional of CLE(4) and conversely CLE(4) can be made sense as a functional of the GFF. This is joint work with Scott Sheffield.

Time:Monday November 22, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Renming Song (University of Illinois, Urbana-Champaign)

Title:MINIMAL THINNESS FOR SUBORDINATE BROWNIAN MOTION IN HALF SPACE

Abstract:In this talk I will present some recent results on the minimal thinness in the half-space for a large class of subordinate Brownian motions, including symmetric stable processes and sums of Brownian motion and independent stable processes. The main result is that the same test for the minimal thinness of a subset of $H$ below the graph of a non-negative Lipschitz function is valid for all processes in the considered class. This talk is based on a joint paper with Panki Kim and Zoran Vondracek.

Time:Monday November 15, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Omer Angel (University of British Columbia)

Title:A PHASE TRANSITION FOR DYADIC TILINGS

Abstract:A dyadic tile in the unit square $[0,1]^2$ is a rectangle of the form $[i/2^a,(i+1)/2^a] \times [j/2^b,(j+1)/2^b]$ for some $i,j,a,b$. Such a tile is of order $a+b$. Suppose each tile of order $n$ is available with some probability $p$. Is it possible to tile the square using a subset of $2^n$ of the available tiles? We show that this is possible with high probability if and only if $p>p_c$ for some $0

Time:Monday November 8, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Gregory Lawler (University of Chicago)

Title:SCHRAMM-LOEWNER EVOLUTION IN MULTIPLY CONNECTED DOMAINS

Abstract:The chordal Schramm-Loewner evolution (SLE) is a measure on curves connecting boundary points of a domain. Schramm defined the process for simply connected domains, but it is not immediate how to extend the definition to multiply connected domains. I will discuss an approach using the Brownian loop measure and some work in progress showing that the measure is well defined. The recent work uses ideas from a recent paper of Dapeng Zhan on reversibility of whole plane SLE.

Time:Monday November 1 , 2010 at 2:30 pm.

Location: SMI 211

Speaker:Russ Lyons (Indiana University)

Title:UNIFORM SPANNING FORESTS, THE FIRST $\ell^2$-BETTI NUMBER, AND UNIFORM ISOPERIMETRIC INEQUALITIES

Abstract:Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. It turns out that they are related to the first $\ell^2$-Betti numbers of groups. We illustrate this by proving a uniform isoperimetric inequality for Cayley graphs.

Time:Monday October 25, 2010 at 2:30 pm.

Location: SMI 211

Speaker:Dan Romik (University of California, Davis)

Title:RCTIC CIRCLES, RANDOM DOMINO TILINGS AND SQUARE YOUNG TABLEAUX

Abstract:It is well-known that domino tilings and certain other random combinatorial or \break statistical-physics models on a two-dimensional lattice exhibit a spatial phase transition between an "arctic" or "frozen" region where the behavior of the random object is asymptotically deterministic and a "temperate" region where truly random behavior is observed. One famous example of such a phenomenon is the Arctic Circle Theorem due to Jockusch, Propp and Shor, which shows that for uniformly random domino tilings of a particular region known as the Aztec Diamond, the curve that forms the interface between the frozen and temperate regions converges to a circle. Cohn, Elkies and Propp later derived a more detailed result about the limiting height function of the typical domino tiling of the Aztec diamond. In this talk, I will present a new proof of the Cohn-Elkies-Propp limit shape result for the height function based on a connection to alternating-sign matrices and a variational analysis. The proof highlights a surprising connection of this result to another arctic-circle type phenomenon observed in a different problem involving uniformly random square Young tableaux.

Time:Monday October 18, 2010 at 2:30 pm.

Location: MEB 245

Speaker:Philip Matchett Wood (Stanford University)

Title:RANDOM TRIDIAGONAL DOUBLY STOCHASTIC MATRICES

Abstract:Let $T_n$ be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally as birth and death chains with a uniform stationary distribution. One can think of a ~Qtypical~R matrix $T_n$ as one chosen uniformly at random, and this talk will present a simple algorithm to sample uniformly in $T_n$. Once we have our hands on a 'typical' element of $T_n$, there are many natural questions to ask: What are the eigenvalues? What is the mixing time? What is the distribution of the entries? This talk will explore these and other questions, with a focus on whether a random element of $T_n$ exhibits a cutoff in its approach to stationarity. Joint work with Persi Diaconis.

Time:Wednesday October 13 , 2010 at 2:30 pm.

Location: GUG 204

Speaker:Jean-Francois Le Gall (Universite Paris-Sud, Orsay)

Title:RANDOM RECURSIVE TRIANGULATIONS OF THE DISC AND FRAGMENTATION THEORY

Abstract:We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension $(\sqrt{17}-1)/2$ and that it can be described as the geodesic lamination coded by a H\"older continuous random continuous function. We also discuss recursive constructions of triangulations of the $n$-gon that give rise to the same continuous limit when $n$ tends to infinity. Proofs rely on asymptotic results from fragmentation theory. Joint work with Nicolas Curien.

Time:Monday October 4, 2010 at 2:30 pm.

Location: MEB 245

Speaker:Tonci Antunovic (University of California, Berkeley)

Title:ISOLATED ZEROS FOR BROWNIAN MOTION WITH VARIABLE DRIFT

Abstract:It is well known that standard one-dimensional Brownian motion B(t) has no isolated zeros almost surely. We show that for any alpha<1/2 there are alpha-Hölder continuous functions f(t) for which the process B(t)-f(t) has isolated zeros with positive probability. We also prove that for any f(t), the zero set of B(t)-f(t) has Hausdorff dimension at least 1/2 with positive probability, and 1/2 is an upper bound if f(t) is 1/2-Hölder continuous or of bounded variation. This is a joint work with Krzysztof Burdzy, Yuval Peres and Julia Ruscher.

Time:Monday May 10, 2010 at 2:30 pm.

Location: RAI 116

Speaker:Svante Janson (Uppsula)

Title:Bootstrap percolation on random graphs.

Abstract:Bootstrap percolation on a graph is a process where an "infection" is spread such that a vertex gets infected if at least 2 of its neighbors are infected (the number 2 may be replaced by another constant threshold). (This is hardly a realistic model for ordinary infectious diseases, but maybe for rumors and beliefs.) Bootstrap percolation has been studied on both deterministic and random graphs. I consider it on the random graph G(n,p), with a number a of initially infected nodes. I will present a simple method to analyze this process, and in particular to find for which p and a the infection will spread to all or almost all vertices. (Joint work with Tomasz Luczak, Tatyana Turova and Thomas Vallier)

Time:Monday May 3, 2010 at 2:30 pm.

Location: RAI 116

Speaker:James Lee (UW)

Title:Diffusion on transitive graphs

Abstract:We prove that on any infinite, connected, transitive graph, the probability for the simple random walk to be within eps*sqrt{t} of the origin is O(eps). A similar estimate holds for finite graphs, up to the relaxation time of the walk.

Our approach uses non-constant equivariant harmonic mappings to reduce the problem to a study of L^2-valued martingales, a setting where we can apply Fourier-analytic methods. For the case of discrete, amenable groups, we give a new construction of such harmonic mappings based on the heat flow from a Folner set.

Joint work with Yuval Peres.

Time:Monday April 26, 2010 at 2:30 pm.

Location: RAI 116

Speaker:Sebastian Andres (UBC)

Title:Particle Approximation of the Wasserstein Diffusion

Abstract:In this talk we present a finite dimensional approximation of the Wasserstein diffusion on the unit interval constructed by Sturm and von Renesse. More precisely, the empirical measure process associated to a system of interacting, two-sided Bessel processes with dimension 0 < ? < 1 converges in distribution to the Wasserstein diffusion under the equilibrium fluctuation scaling. The passage to the limit is based on Mosco convergence of the associated Dirichlet forms in the generalized sense of Kuwae/Shioya, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. In analogy to the classical Bessel process the approximating system fails to be a semi-martingale, but it satisfies the Feller property, which can be proven by using the theory of Sobolev spaces with Muckenhoupt weights. This is joint work with Max von Renesse.

Time:Monday April 19, 2010 at 2:30 pm.

Location: RAI 116

Speaker:Brigitta Vermesi (IPAM)

Title:Can Brownian motion with drift be space-filling?

Abstract:It is well known that, for d>1, d-dimensional Brownian motion does not hit points almost surely. How about Brownian motion with drift? For functions f(t) in the Dirichlet space D[0,1], the path B(t)-f(t) has the same almost sure properties as a Brownian path, hence B-f does not hit points. In this talk, we will construct a continuous function f(t) for which B-f not only does it hit points, but its range covers an open set almost surely. This is joint work with Tonci Antunovic and Yuval Peres.

Time:Friday, April 9, 2010 at 2:30 pm.

Location:SMI 107

Speaker:Ed Waymire (Oregon State)

Title:Tree Polymers: Some Recent Results and Problems

Abstract:Tree polymers are simplifications of 1+1 dimensional lattice polymers made up of polygonal paths of a (nonrecombining) binary tree having random path probabilities. As in the case of lattice polymers, the path probabilities are (normalized) products of i.i.d. positive random weights. The a.s. probability laws of these paths are of interest under weak and strong types of disorder. The case of no disorder provides a benchmark since the polymers are simple symmetric random walk paths where all of the probability laws are known. We will discuss some recent results, speculation and open problems for this class of models. This is largely based on joint work with Stan Williams.

Time:Monday April 5, 2010 at 2:30 pm.

Location:SMI 120

Speaker:Tiefeng Jiang (University of Minnesota)

Title:Moments of Traces for Circular Beta-ensembles

Abstract:Let x_1, ..., x_n be random variables from Dyson's circular beta-ensemble with probability density function prod_{1 leq j< k leq n} |e^{i x_j} - e^{i x_k}|^{beta}. For each n geq 2 and beta>0, we obtain inequalities on expectations of p_{mu}(Z_n), where Z_n=(e^{i x_1}, \cdots, e^{i x_n}) and p_{mu} is the power-sum symmetric function for partition mu. When beta=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have limit results on the moments. These results apply to the three important ensembles: Circular Orthogonal Ensemble (beta=1), Circular Unitary Ensemble (beta=2) and Circular Symplectic Ensemble (beta=4). The main tool is the Jack function. This is a joint work with Sho Matsumoto.

Time:Monday March 8, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Jean-Dominique Deuschel (Technische Universitat Berlin)

Title:Gradient Gibbs distribution with non-convex potential at high temperature

Abstract:We consider a gradient Gibbs measure with non convex potential and show that it behaves at high temperature like a gaussian free field. The proof is based on the fact that the marginal distribution of the even sites has a stictly convex Hamiltonian for which we can apply the random walk representation.

This is a joint work with Codina Cotar

Time:Friday March 5, 2010 at 2:30 pm.

Location:PDL C36

Speaker:Michael Kozdron (University of Regina, Canada)

Title:A rate of convergence for loop-erased random walk to SLE(2)

Abstract:Among the open problems for SLE suggested by Oded Schramm in his 2006 ICM talk is that of obtaining reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE. In this talk we derive a rate for the convergence of the Loewner driving function for loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2).

This talk is based on joint work with Christian Benes (CUNY) and Fredrik Johansson (KTH).

Time:Monday March 1, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Perla Sousi (University of Cambridge)

Title:Collisions of random walks

Abstract:Regarding his 1920 paper proving recurrence of random walks in Z^2, Polya wrote that his motivation was to determine whether 2 independent random walks in Z^2 meet infinitely often. Of course, in this case, the problem reduces to the recurrence of a single random walk in Z^2, by taking differences. Perhaps surprisingly, however, there exist graphs G where a single random walk is recurrent, yet G has the finite collision property: two independent random walks in G collide only finitely many times almost surely. Some examples were constructed by Krishnapur and Peres (2004), who asked whether critical Galton-Watson trees conditioned on nonextinction also have this property. In this talk I will answer this question as part of a systematic study of the finite collision property. In particular, for two classes of graphs, wedge combs and spherically symmetric trees, we exhibit a phase transition for the finite collision property when growth parameters are varied. I will state the main theorems and give some ideas of the proofs.

This is joint work with Martin Barlow and Yuval Peres.

Time:Monday February 22, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Asaf Nachmias (Microsoft Research)

Title:Percolation on groups

Abstract:In the past decade there has been much activity on percolation on Cayley graphs of groups. The emphasis in this line of research is the interplay between geometric properties of the group (such as amenability, hyperbolicity, number of ends, etc.) and the behavior of percolation on its Cayley graph.

We will survey key theorems and questions of this field, and present a result obtained jointly with Itai Benjamini and Yuval Peres: the critical percolation probability p_c of a non-amenable graph with high girth is close to the its value on an infinite regular tree. This is a particular case of a conjecture due to Schramm on the locality of p_c.

Time:Monday February 8, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Yanxia Ren (Peking University, China)

Title:Traveling Wave Solutions of Nonlinear Differential Equations Associated to Super-Brownian Motions

Abstract:We consider the problem of existences, uniqueness, and asymptotics of the traveling wave solution of a parabolic nonlinear differential equation associated to super-Brownian motion with general branching mechanism. Probabilistic methods are successfully used to solve the basic problems about the traveling wave solutions of nonlinear differential equations associated to branching Brownian motion (see, for example, Kyprianou Ann. I. H. Poincoi\'{e}-PR, 2004) and references therein). We construct analogous arguments for super-Brownian motion using Dynkin's exit measures, martingales and spine decomposition of super-Brownian motion after a martingale change of measure.

The talk is based on a joint work in progress with A.E. Kyprianou, R.-L. Liu and A. Murillo-Salas.

Time:Monday February 1, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Kyeong-Hun Kim (Korea University, South Korea)

Title:An L^{2}-theory of stochastic PDEs driven by Lévy processes

Abstract:Stochastic PDEs are used to describe many physical and biological systems influenced by some random noises. In this talk, we consider a class of stochastic PDEs driven by Lévy processes, and we present the uniqueness and existence results of pathwise solution in L^{2}-spaces. Some fundamental estimates for L^{p}-theory, where p>2, will also be discussed.

This talk is based on a joint work with Z.-Q. Chen.

Time:Monday January 25, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Remco van der Hofstad (Eindhoven University of Technology, the Netherlands)

Title:Critical behavior in inhomogeneous random graphs

Abstract:Empirical findings have shown that many real-world networks share fascinating features. Many real-world networks are small-worlds, in the sense that typical distances are much smaller than the size of the network. Further, many real-world networks are scale-free in the sense that there is a high variability in the number of connections of the elements of the networks. Therefore, such networks are highly inhomogeneous.

Spurred by these empirical findings, models have been proposed for such networks. In this talk, we shall discuss a particular class of random graphs, in which edges are present independently but with unequal edge occupation probabilities that are moderated by appropriate vertex weights. For such models, it is known precisely when there is a giant component, meaning that a positive proportion of the vertices is connected to one another.

We discuss what happens precisely at criticality, a problem having strong connections to statistical mechanics. We identify the scaling limit of the connected components ordered by their size. Our results show that, for inhomogeneous random graphs with highly variable vertex degrees, the critical behavior admits a transition when the third moment of the degrees turns from finite to infinite. When the third moment is finite, the largest clusters in a graph of n vertices have size n^{2/3} and the scaling limit equals that on the Erdoes-Renyi random graph. When the third moment is infinite, the largest clusters have size to n^{\alpha} for some \alpha\in (1/2,2/3), and the scaling limit is rather different. Similar phase transitions have been shown to occur for typical distances in such random graphs when the variance of the degrees turns from finite to infinite.

[This is joint work with Shankar Bhamidi and Johan van Leeuwaarden.]

Time:Wednesday January 20, 2010 at 2:30 pm.

Location:SMI 311

Speaker:Anne Fey (Technische Universiteit Delft, The Netherlands)

Title:Sandpiles, Staircases and Self-organized criticality

Abstract:In the fixed-energy sandpile model, `activity density' jumps are observed as the particle density increases, in some cases even seeming to form a devil's staircase. After presenting new results on this topic, we will focus on the first jump, from zero to nonzero activity density. The particle density where this transition takes place, plays a role in a popular heuristic argument to explain self-organized criticality in sandpile models: it is argued that another, differently defined particle density is in fact the same. This conjecture was supported by simulations and generally believed to be true. However, we can now for several cases exactly calculate both densities, giving very close, but unequal values.

Joint work with Lionel Levine and David Wilson.

Time:Monday January 11, 2010 at 2:30 pm.

Location:MEB 235

Speaker:Panki Kim (Seoul National University and University of Illinois at Urbana-Champaign)

Title:Global Heat Kernel Estimates for Symmetric Jump Processes

Abstract:In this talk, we discuss the behavior of heat kernel for symmetric jump-type process with jumping kernels comparable to radially symmetric function on the Euclidean space. Sharp two-sided heat kernel estimates for both small and large time will be discussed.

This is a joint work with Zhen-Qing Chen and Takashi Kumagai.

Time:Monday December 7, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Soumik Pal (University of Washington)

Title:SPECTRAL PROPERTIES OF LARGE REGULAR RANDOM GRAPHS

Abstract:Regular random graphs are important models in probabilistic combinatorics and have sustained interest for decades.

Recent theory, spurred in part by the study of Expanders, have focussed on spectral properties of the adjacency matrices of a sequence of regular random graphs with a fixed degree and a growing number of vertices (order). A striking example in this vein is the celebrated proof of the Alon Conjecture by Joel Friedman. We will talk about the case when the degree also grows, albeit slowly, with the order. Although the graphs remain sparse, their spectral properties begin to resemble those of the Gaussian Orthogonal Ensemble, or real Wigner matrices. This can be seen in the convergence of the empirical spectral distribution to the semicircular law and theorems that indicate that their eigenvectors are approximately uniformly distributed over the sphere. Thus, this can be seen as an attempt to push ``universality'' in the classical Random Matrix Theory context to sparse, non-Wigner matrices. We will talk about theorems, tantalizing unproven conjectures, and some open problems. Our methods are a combination of analytical tools such as Stieltjes transforms and combinatorial ideas such as local tree approximation. This is based on joint work with Ioana Dumitriu.

Time:Monday November 30, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Thinh Nguyen (Vietnam National University)

Title:ON SOME PROBLEMS OF RANDOM BOUNDED OPERATORS

Abstract:A random operator from a space $X$ into a space $Y$ is a linear continuous mapping which assign to each element $x$ in $X$ (input) some element $Ax$ which is not known exactly, or in the other words, $Ax$ is a random element in $Y$ (output). So random operators can be regarded as a random generalization of deterministic linear operators. It is desirable to obtain results parallel to those well-known in theory of linear operator for random operator. In this talk we will recommend the notion of random bounded operations, the failure of Banach-Steinhaus version for random bounded operations, the problem of extending and the spectral theorem for random bounded operations and related problems.

Time:Monday November 23, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Julia Ruscher (Microsoft Research)

Title:PROPERTIES OF SUPERPROCESSES

Abstract:A large class of random spatial processes involves populations that undergo reproduction (birth and death happen randomly according to a certain mechanism) in addition to random spatial motion (e.g. Brownian motion). Superprocesses are exotic Markov processes in that sense that they are the scaling limits of many such processes. We will introduce classes of Superprocesses and look at their relationship and properties, in particular the Hausdorff dimension of the support of Superprocesses.

Time:Monday November 16, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Mauricio Duarte (University of Washington)

Title:DIRICHLET FORMS AND REFLECTING BROWNIAN MOTION

Abstract:The theory of Dirichlet forms is yet another example of the rich interplay between Analysis and Probability. The probabilistic part, initiated by the fundamental work of Fukushima in the early 70's, connects symmetric Markov processes and Dirichlet forms in a very powerful way. More recently, the connection has been extended to non-symmetric forms. This extension makes possible the characterization of all forms that are related to `nice' Markov processes, under the notion of quasi-regularity.

In this talk, we will show how to construct a Markov process out of a Dirichlet form, and as an application, we will show a construction of Brownian motion with oblique reflection in a smooth domain.

Time:Monday November 9, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Richard Bass (University of Connecticut)

Title:STABLE-LIKE PROCESSES

Abstract:The class of stable-like processes is a subset of the class of multidimensional jump processes. They stand in the same relationship to stable processes as multidimensional diffusions do to Brownian motion. I'll describe several models of stable-like processes, and then talk about relatively recent results, such as uniqueness of martingale problems and Harnack inequalities. Then I'll talk about very recent results on the regularity of potentials of stable-like processes.

Time:Monday November 2, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Geoffrey Grimmett (Cambridge University and Microsoft)

Title:CLASSICAL REPRESENTATIONS FOR THE QUANTUM ISING MODEL

Abstract:The quantum Ising model in d dimensions may be mapped to a (d+1)-dimensional `continuous' Ising model of classical type. This may be solved using an approach developed by Aizenman and others under the name `random-current representation'. We prove the sharpness of the phase transition, and establish two inequalities for critical exponents. The value of the ground-state critical point may be calculated rigorously in one dimension, and the corresponding transition is continuous. (Joint work with Jakob Björnberg.)

Time:Friday October 23, 2009 at 2:30 pm. (Colloquium Talk)

Location:Mary Gates Hall 241

Speaker:Persi Diaconis (Stanford University)

Title:Shuffling Cards and Adding Numbers

Abstract:When several large integers are added in the usual way 'carries' occur along the way. It is natural to ask: 'About how many carries are there and how are they distributed for typical numbers?' It turns out that these questions are intimately related to the mathematics of the usual way we shuffle cards. I will explain the mathematics of 'carries' (they are cocycles!), shuffling and the connection. This is joint work with Jason Fulman.

Time:Thursday October 22, 2009 at 1:30 pm. (Combinatorics Seminar)

Location:SMI 205

Speaker:Persi Diaconis (Stanford University)

Title:Random Walks and Hyperplane Arrangements

Abstract:

Time:Monday October 19, 2009 at 2:30 pm.

Location:MEB 235

ipeaker:Brigitta Vermesi (IPAM)

Title:INTERSECTION EXPONENTS FOR BIASED RANDOM WALKS ON CYLINDERS

Abstract:In the past decade, there have been significant advances in the study of 2-dimensional critical systems in statistical physics, in particular due to the introduction of Schramm Loewner Evolution (SLE). For example, some critical exponents for planar Brownian motion have been computed exactly using SLE. But what can we say about the same exponents in the case of 3-dimensional Brownian motion? We approach the question by looking at a simpler model: biased random walks on d-dimensional cylinders. In this talk, I will describe the random walk problem and explain how it relates to the 3-dimensional Brownian motion case. This leads to a conjecture about exponents for Brownian motion.

Time:Monday October 12, 2009 at 2:30 pm.

Location:MEB 235

Speaker:Allan Sly (Microsoft Research)

Title:MIXING IN TIME AND SPACE OF GIBBS MEASURES

Abstract:The mixing time of the Glauber dynamics is often closely related to the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions are of interest in probability, statistical physics and theoretical computer science. I will discuss some recent progress in understanding the mixing time of the Glauber dynamics for the Ising model and for random colorings of random graphs.

Time:Wednesday October 7, 2009 at 3:45pm. (Joint with Differential Geometry/PDE Seminar)

Location:PDL C-36

Speaker:Elton Hsu (Northwestern University)

Title:Volume Growth, Brownian motion, and Stochastic Completeness of a Complete Riemannian manifold

Abstract:A geodesically complete Riemannian manifold is called stochastically complete if its heat kernel (the minimal fundamental solution of the parabolic Laplace-Beltrami operator) is integrated to one. Since the heat kernel is the transition density function of Riemannian Brownian motion, a manifold is stochastically complete if and only if Brownian motion does not explode. To find a proper geometric condition for stochastic completeness is an old geometric problem. The first result in this direction was due to S. T. Yau, who proved that a Riemannian manifold is stochastically complete if its Ricci curvature is bounded from below by a constant. It has been know for quite some time that the property of stochastic completeness is intimately related to the volume growth of a Riemannian manifold. We study stochastic completeness by looking at the more refined question of upper escaping rates of Riemannian Brownian motion. We show how the Neumann heat kernel, time reversal of reflecting Brownian motion, and volumes of geodesic balls come together and give an elegant and often sharp upper bound of the escaping rate solely in terms of the volume growth function without any extra geometric restriction.

This is a joint work with Guang Nan Qin of Institute of Applied Mathematics of the Chinese Academy of Sciences.

Time:Friday June 5, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Asaf Nachmias (Microsoft Research)

Title:The one-arm exponent in high-dimensional percolation

Abstract:We study the probability that the origin is connected to the sphere of radius R in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. I will present highlights of the proof that this probability decays like R^{-2}.

Joint work with Gady Kozma.

Time:Friday May 29, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Nathaniel Blair-Stahn (University of Washington)

Title:Survival and limiting configurations in the two-type Richardson model

Abstract:The two-type Richardson model is an interacting particle system simulating two randomly spreading species competing for space in two or more dimensions. The main question of interest is whether there is a positive probability that both species survive indefinitely. That is, can both species eventually conquer an infinite region, or does one species end up completely surrounded by the other one almost surely?

The model can be described in terms of two underlying first-passage percolation processes, allowing an analysis based on subadditive ergodic theory. In particular, the so-called 'shape theorem' implies that the model should behave in an essentially deterministic way on large scales. The behavior of this limiting deterministic process thus gives a prediction for the behavior of the random process. Using this idea, I will explain how to analyze the behavior of the two-type Richardson model when one species initially occupies an infinite region, and how this analysis provides information about the growth of the process started from a finite initial configuration.

Time:Friday May 22, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Jian Ding (UC Berkerley)

Title:Total variation cutoff in birth-and-death chains

Abstract:The *cutoff phenomenon* describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time birth-and-death chains, started at an endpoint, with convergence measured in *separation*. It is natural to ask whether the conjecture holds for these chains in the more widely used *total-variation* distance.

In this work, we confirm the above conjecture for all continuous-time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time.

In addition, we show that for any lazy (or continuous-time) birth-and-death chain with stationary distribution $\pi$, the separation $1 - p^t(x,y)/\pi(y)$ is maximized when $x,y$ are the endpoints. Together with the above results, this implies that total-variation cutoff is equivalent to separation cutoff in any family of such chains.

Joint work with Eyal Lubetzky and Yuval Peres.

Time:Friday May 15, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Krzysztof Burdzy (University of Washington)

Title:The future of solar energy

Abstract:Can one design a rough mirror that reflects the light in any prescribed direction? The answer is "no" at this level of generality but it turns out that the answer is "yes" for a very large and "natural" class of reflection angles. I will present the original motivation, the original problem, and a tentative theorem. I will also point out how this mathematical project may revolutionize solar energy production.

This is work in progress, joint with Omer Angel and Scott Sheffield.

Time:Friday May 8, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Ron Peled (Courant Institute)

Title:Phase Transitions in Gravitational Allocation

Abstract:Given a Poisson point process of unit masses ("stars") in dimension d >= 3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. The allocation is translation equivalent - the shape of cells do not depend on absolute position in space. We investigate the quantitative geometry of the allocation's cells. Our first result shows that a.s. all cells are bounded and that their diameters have exponential tails. We continue and investigate large deviations for the cells. We find that the probability that mass exp(~H~RR^t) in a cell travels distance R decays like exp(~H~RR^(f_d(t))) and we identify the functions f_d exactly. These functions are piecewise linear and the discontinuities of f'_d represent phase transitions. We observe two distinct behaviors: In dimension 3, large deviations are due to a "distant attracting galaxy" which attracts the mass from afar. In dimensions d >= 5, large deviations are due to a "wormhole". A thin tube along which the star density increases monotonically and which pulls the mass through it. In dimension 4 we have a double phase transition with a transition between low- dimensional behavior (attracting galaxy) and high-dimensional behavior (wormhole) at t = 4/3. As consequences, we determine the tail behavior of the distance from a star to a uniform point in its cell, and prove a sharp lower bound for the tail probability of the cell's diameter, matching our earlier upper bound.

This is joint work with Sourav Chatterjee, Yuval Peres and Dan Romik.

Time:Friday May 1, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Soumik Pal (University of Washington)

Title:Interacting diffusion models of capital markets

Abstract:Consider a multidimensional interacting diffusion model where the drift and the diffusion coefficients for individual coordinates are dependent on the relative sizes of their current values compared to the others. The coordinate processes are otherwise exchangeable. Two such models were introduced by Fernholz and Karatzas as models for equity markets where it has long been empirically observed that the rate of growth and the fluctuation in the value of a stock capital depends on its relative value with respect to the entire market. We consider one of the models, named the volatility-stabilized market model, where the drift and the diffusion parameters are functions of the ratio of the current value to the total sum over all the coordinates. We show that these models are deeply connected to squared-Bessel processes and the multi-allele Wright-Fisher diffusions in genetics. This allows us to derive explicit formulas regarding the behavior of these processes at fixed and stopping times. As side results we derive a novel symmetry of the Bessel processes and produce a new proof of the transition density of the Wright-Fisher diffusions, which was originally derived by Griffiths by using bi-orthogonal polynomial expansions.

Time:Tuesday April 21, 2009 at 2:30 pm.

Location:Condon Hall (CDH) 135

Speaker:Daniel Conus (University of Utah)

Title:The non-linear wave equation in high dimensions: existence, Holder-continuity and Ito-Taylor expansion

Abstract:The main topic of this talk is the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time.

In dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution. Hence, we first develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation.

With this extended stochastic integral, we establish existence of a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specific class of processes.

In the case of affine multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p \geq 1). From this, we deduce Holder-continuity of the solution. The Holder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated Ito-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem.

(Joint work with Robert C. Dalang, Swiss Federal Institute of Technology)

Time:Friday, April 17, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Zhen-Qing Chen (University of Washington)

Title:Symmetric Markov Processes and Heat Kernel Estimates

Abstract:In this talk, I will describe recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric discontinuous processes (or equivalently, a class of symmetric integro-differential operators). A prototype of the Markov processes under consideration is the mixture of symmetric diffusion of uniformly elliptic divergence form operator and symmetric stable-like processes on $R^d$.

We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Holder estimate and parabolic Harnack inequalities for their parabolic functions. To establish these results, we employ methods from both probability theory and analysis.

Joint work with Takashi Kumagai.

Time:Friday, April 10, 2009 at 2:30 pm.

Location:Loew Hall 101

Speaker:Ping Ao (University of Washington and Shanghai Jiao Tong University)

Title:New Approach to Stochastic Differential Equations in Higher Dimensions

Abstract:A new way to treat stochastic differential equations emerged from our biological studies will be discussed. It is different from both Ito and Stratonovich. Two of the prominent features are that, it can be applied to situations without detailed balance (time irreversibility, or other terms depending on one's preference), and, the final stationary distribution, if exists, is exactly of the Boltzmann-Gibbs distribution type. With the help of Feynman-Kac formulae several novel stochastic dynamical equalities are also suggested. Such formulation provides a consistent foundation to derive statistical mechanics and thermodynamics.

References:

- P. Ao, Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics. Communications in Theoretical Physics 49 (2008) 1073-1090.

http://ctp.itp.ac.cn/qikan/Epaper/zhaiyao.asp?bsid=2817 ;- P. Ao, Global view of bionetwork dynamics: adaptive landscape. Journal of Genetics and Genomics 36 (2009) 63-73.

doi:10.1016/S1673-8527(08)60093-4 ;- L. Yin and P. Ao, Existence and Construction of Dynamical Potential in Nonequilibrium Processes without Detailed Balance. Journal of Physics A39 (2006) 8593-8601.

http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.0379v1.pdf ;- C. Kwon, P. Ao and D.J. Thouless, Structure of Stochastic Dynamics near Fixed Points. Proc. Nat'l Acad. Sci. (USA) 102 (2005) 13029-13033.

http://www.pnas.org/content/102/37/13029.full.pdf+html ;- P. Ao, Potential in Stochastic Differential Equations: Novel Construction. J. Phys. A37 L25-L30 (2004).

http://www.iop.org/EJ/abstract/0305-4470/37/3/L01/ ;- P. Ao, Boltzmann-Gibbs Distribution of Fortune and Broken Time-Reversible Symmetry in Econodynamics. Communications in Nonlinear Sciences and Numerical Simulation 12 (2007) 619-626.

http://dx.doi.org/10.1016/j.cnsns.2005.07.004

Time:Thursday, March 19, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Zbigniew J. Jurek (University of Wroclaw, Poland)

Title:GENERALIZED LEVY STOCHASTIC AREAS AND SELFDECOMPOSABILITY PROPERTY

Abstract:For a planar Brownian motion $B_t = ( Z_t,\tilde{Z}_t)$ and a stochastic area process $$ A_u = \int_{0}^{u} Z_sd\tilde{Z}_s - \tilde{Z}_sdZ_s , \qquad u>0, $$ Paul L\'evy (1950) proved that its conditional characteristic function is of the form $$ E[e^{it {A}_u} | {B}_u=a] = {tu\over \sinh tu}\, \exp[- {|a|^2\over 2u}(tu \coth tu-1)],\ \ t\in R, $$ where $a \in R^2$ and $u \ge 0$ are fixed. Thus, in particular, $$ {E}[e^{it {A}_u} | {B}_u=(\sqrt{u},\sqrt{u} )] = {tu\over \sinh tu}\,\,\cdot \, \exp[-(tu \coth tu -1)],\ \ t\in R. \eqno(1) $$ We will show that the two factors in (1) are intimately related. Namely the first is a characteristic function of a selfdecomposable distribution while the second one is its ``background driving probability measure''.

Reference: {{Stat. \& Probab. Letters}, vol. {64}(2003), pp. 213-222.

Time:Monday, March 9, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Soumik Pal (University of Washington)

Title:A combinatorial analysis of interacting diffusions

Abstract:We consider a particular class of multidimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant over cones. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the n-dimensional space and can be extremely resistant to a more detailed understanding. One can see that the structure of the invariant probability measure is intertwined with the geometry of the cones where the drift function is a constant. In this article we pursue results that make this connection precise. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. As broad classes of examples we analyze interactions defined by Coxeter groups actions and weighted graphs by this method.

Time:Monday, March 2, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Ryan Card (University of Washington)

Title:Brownian Motion with Boundary Diffusion

Abstract:Diffusion in a domain can be described by a second order differential operator plus a boundary condition. Dirichlet and Neumann boundary conditions are familiar to us, which corresponds to killed and reflected processes respectively. General class of boundary conditions was found by Wentzell in 1959. I will describe how processes with Wentzell's boundary condition behave, and present an invariance principle for a class of Markov processes that converge to diffusion with Wentzell's boundary conditions. Time permitting, I will talk about some curious properties of these processes and how they can be used to prove Boundary Harnack Principle for harmonic functions that satisfy boundary diffusive condition.

Time:Monday, February 23, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Amites Sarkar (Western Washington University)

Title:Partitioning Random Geometric Covers

Abstract:I'll present some new results on partitioning both random and non-random geometric covers. For the random results, let P be a Poisson process of intensity one in the infinite plane R^2, and surround each point x of P by the open disc of radius r centered at x. Now let S_n be a fixed disc of area n>>r^2, and let C_r(n) be the set of discs which intersect S_n. Write E_r^k for the event that C_r(n) is a k-cover of S_n, and F_r^k for the event that C_r(n) may be partitioned into k disjoint single covers of S_n. I'll sketch a proof of the inequality Prob(E^k_r | F^k_r) \leq c_k/log n, which is best possible up to a constant. The non-random result is a classification theorem for covers of R^2 with half-planes that cannot be partitioned into two single covers. It was motivated by a desire to understand the obstructions to k-partitionability in the original random context. This is all joint work with Paul Balister, Bela Bollobas and Mark Walters.

Time:Tuesday, February 10, 2009 at 1:30 pm.

Location:Gates Commons, 6th floor of CSE building

Speaker:Yuval Peres (Microsoft Research)

Title:The Unreasonable Effectiveness of Martingales

Abstract:I will illustrate the effectiveness of Martingales and stopping times with four examples, involving waiting times for patterns in coin tossing, random graphs, mixing of random walks, and metric space embedding. A common theme is the way stopping time arguments often circumvent the wasteful "union bound" (bounding of the probability of a union by the sum of the individual probabilities).Note Special Time and Place

Time:Monday, February 9, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Matt Kahle (Stanford University)

Title:Random geometric simplicial complexes

Abstract:Choose n random points, independent and identically distributed in R^d, connect them if they are within distance r, and build a simplicial complex on this graph. (We will define and discuss two of the most standard geometric constructions: the Rips and Cech complexes.) What is the topology of this complex likely to be like, and how does it change as r grows from 0 to infinity? We are able to find a fairly detailed picture. In some cases we are able to prove Central Limit Theorems for the dimensions of certain homology groups. One interesting feature is that the underlying topological properties which we study are intrinsically non-monotone.

Time:Monday, February 2, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Panki Kim (Seoul National University)

Title:Potential theory of 1-dimensional subordinate Brownian motions with continuous components

Abstract:Suppose that S is a subordinator (1-dimensional increasing Levy process) with a nonzero drift and W is an independent 1-dimensional Brownian motion. X_t = W(S_t) is called the subordinate Brownian motion. In this talk, we discuss sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and a boundary Harnack principle. This is a joint work with Renming Song and Zoran Vondracek.

Time:Monday, January 26, 2009 at 2:30 pm.

Location:MGH 085

Speaker:Ioana Dumitriu (University of Washington)

Title:In search of the elusive beta: from full to tridiagonal random matrices

Abstract:For some classical random matrix ensembles, eigenvalue distributions are computable exactly. When changing the field of entries (from real to complex and to quaternion), these distributions change little, and in very simple ways, according to a $\beta$ parameter. This creates "interpolating" ensembles of eigenvalues with no full matrix model. If one lets go of the fullness requirement, tridiagonal models can be constructed which correspond to every $\beta$. These models are great in many ways, and insufficient in others. I will speak about constructing them, as well as about the pros and cons.

Time:Friday, December 5, 2008 at 2:30 pm.

Location:EEB 031

Speaker:Bjoern Kjos-Hanssen (University of Hawaii)

Title:INFINITE SUBSETS OF RANDOM SETS OF NATURAL NUMBERS

Abstract:A Martin-L\"of random set of natural numbers is one that, considered as an infinite binary sequence, satisfies the Law of Large Numbers and all other ``effective'' laws. This is made precise via Turing computability. Using results of J. Hawkes and R. Lyons on intersection probabilities for Galton-Watson trees, we show that there exists an infinite subset of a Martin-L\"of random set that does not compute any Martin-L\"of random set. (In other words: a photon detector that rarely absorbs incoming photons is generally not a useful randomness source.) It is not known how dense such a subset can be; J. Miller has shown that it can contain at least $n$ numbers smaller than $n^3$ for every $n$.

Time:Thursday, December 4, 2008 at 2:30 pm.

Location:MOR 234

Speaker:Gady Kozma (Weizmann Institute)

Title:LOOP ERASED RANDOM WALK IN 3D

Abstract:Take a simple random walker, and trace its path, removing each loop as it is created. The result is a model for a random simple curve known as loop-erased random walk. Defined in 1980 by Lawler, it's curious symmetries and algorithmic aspects have made it into the most tractable non-Gaussian model of critical phenomena. We will discuss all these, and a proof that it has a scaling limit in 3d.

Time:Monday, December 1, 2008 at 2:30 pm.

Location:MEB 243

Speaker:Ravi Kannan (Microsoft Research, India)

Title:A NEW PROBABILITY INEQUALITY AND CONCENTRATION RESULTS

Abstract:Azuma's inequality proves concentration for a martingale-difference sequence $X_1,X_2,\dots,X_n$ assuming absolute bounds on the $X_i$.

Our main result replaces the absolute bounds by bounds on moments of $X_i$ conditioned on previous $X_j$; in addition, it will work for a more general situation than martingale differences. An important feature is that we will use both bounds for worst-case previous $X_j$ as well as ``typical'' case ones.

Our inequality yields sub-Gaussian tail bounds in many situations (which is ruled out for Burkholder-type inequalities). We use our inequality to derive concentration results for two types of problems : (i) Combinatorial - e.g., Longest Increasing sequence, chromatic number of Random graphs, bin-packing. We match Talagrand's celebrated paper in the first two examples and prove the optimal concentration for the third, settling one of his questions. (ii) Polynomials of independent random variables e.g., number of cliques of a given size in a random graph, where, we obtain some new results.

Time:Monday, November 24, 2008 at 2:30 pm.

Location:MEB 243

Speaker:David Wilson (Microsoft Research)

Title:A SHARP THRESHOLD FOR MINIMUM BOUNDED-DEPTH AND BOUNDED DIAMETER SPANNING TREES AND STEINER TREES IN RANDOM NETWORKS

Abstract:In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to $\zeta(3)=1/1^3+1/2^3+1/3^3+... $ as n goes to infinity. We consider spanning trees constrained to have depth bounded by $k$ from a specified root. We prove that if $k > \log_2 \log n+\omega(1)$, where $\omega(1)$ is any function going to infinity with $n$, then the minimum bounded-depth spanning tree still has weight tending to $\zeta(3)$ as $n \to \infty$, and that if $k < \log_2 \log n$, then the weight is doubly-exponentially large in $\log_2 \log n - k$. It is NP-hard to find the minimum bounded-depth spanning tree, but when $k < \log_2 \log n - \omega(1)$, a simple greedy algorithm is asymptotically optimal, and when $k > \log_2 \log n+\omega(1)$, an algorithm which makes small changes to the minimum (unbounded depth) spanning tree is asymptotically optimal. We prove similar results for minimum bounded-depth Steiner trees, where the tree must connect a specified set of m vertices, and may or may not include other vertices. In particular, when $m = const \cdot n$, if $k > \log_2 \log n+\omega(1)$, the minimum bounded-depth Steiner tree on the complete graph has asymptotically the same weight as the minimum Steiner tree, and if $1 <= k <= \log_2 \log n-\omega(1)$, the weight tends to $(1-2^{-k}) \sqrt{8m/n} [\sqrt{2mn}/2^k]^{1/(2^k-1)}$ in both expectation and probability. The same results hold for minimum bounded-diameter Steiner trees when the diameter bound is $2k$; when the diameter bound is increased from $2k$ to $2k+1$, the minimum Steiner tree weight is reduced by a factor of $2^{1/(2^k-1)}$.

Joint work with Omer Angel and Abie Flaxman.

Time:Monday, November 17 2008 at 2:30 pm.

Location:MEB 243

Speaker:Vladimir Minin (University of Washington)

Title:THE ROLE OF COUNTING PROCESSES IN FITTING AND TESTING PARTIALLY OBSERVED CONTINUOUS-TIME MARKOV CHAIN MODELS

Abstract:Continuous-time Markov chains (CTMCs) serve as basic building blocks for many stochastic models in sciences and engineering. Since the likelihood of CTMC-based models is usually easily computable, likelihood-based inference, Bayesian or frequentist, is a preferred mode of operation in applications involving CTMC inference. I will show that under certain conditions, maximum likelihood estimation is equivalent to optimizing an objective function, defined in terms of CTMC-induced counting processes. This observation leads to a novel, more flexible framework for fitting CTMC models. I will also demonstrate how CTMC-induced counting processes can be used to correct for model misspecification. Using examples from molecular evolutionary biology, I will illustrate the usefulness of CTMC-induced counting processes in scientific applications.

Time:Monday, November 3, 2008 at 2:30 pm.

Location:MEB 243

Speaker:Asaf Nachmias (Microsoft Research)

Title:THE ALEXANDER-ORBACH CONJECTURE HOLDS IN HIGH DIMENSIONS

Abstract:It is known that the simple random walk on the unique infinite cluster of supercritical percolation on $Z^d$ diffuses in the same way it does on the original lattice. In critical percolation, however, the behavior of the random walk changes drastically.

The infinite incipient cluster (IIC) of percolation on $Z^d$ can be thought of as the critical percolation cluster conditioned on being infinite. Alexander and Orbach (1982) conjectured that the spectral dimension of the IIC is 4/3. This means that the probability of an $n$-step random walk to return to its starting point scales like $n^{-2/3}$ (in particular, the walk is recurrent). In this work we prove this conjecture when $d>18$; that is, where the lace-expansion estimates hold.

Joint work with Gady Kozma.

Time:Monday, October 27, 2008 at 2:30 pm.

Location:MEB 243

Speaker:Kyeong-Hun Kim (Korea University and University of Washington)

Title:SOME RESULTS ON STOCHASTIC PDE'S WITH MEASURABLE COEFFICIENTS

Abstract:There is very rich literature for SPDEs with continuous coefficients, but very little is known about SPDEs with only measurable coefficients. In this talk, we present an $L_p$-theory of SPDEs with random coefficients depending also on time and space variables, while all the coefficients are only assumed to be measurable.

Time:Monday, October 20, 2008 at 2:30 pm.

Location:MEB 243

Speaker:Russ Lyons (Indiana University and Microsoft)

Title:RANDOM COMPLEXES VIA TOPOLOGICALLY-INSPIRED DETERMINANTS

Abstract:Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. We present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes. On finite complexes, they relate to (co)homology, while on infinite complexes, they relate to $\ell^2$-Betti numbers. One use is to get uniform isoperimetric inequalities.

(Joint with STATISTICS SEMINAR)

Time:Monday, October 13, 2008 at 3:30 pm.

Location:Smith 304

Speaker:Krzysztof Burdzy (University of Washington)

Title:PHILOSOPHY OF PROBABILITY AND ITS RELATIONSHIP (?) TO STATISTICS

Abstract:I will explain why the frequency statistics has absolutely nothing in common with the frequency philosophy of probability. If time permits, I will explain why the Bayesian statistics has absolutely nothing in common with the subjective philosophy of probability. My presentation will be an unbiased estimator of the truth, with subjective probability 90%.

Time:Monday, October 6, 2008 at 2:30 pm.

Location:MEB 243

Speaker:Ori Gurel-Gurevich (Microsoft Research)

Title:RECURRENCE OF THE SIMPLE RANDOM WALK PATH

Abstract:A simple random walk (SRW) on a graph is a Markov chain whose state space is the vertex set and the next state distribution is uniform among the neighbors of the current state. A graph is called recurrent if a SRW on it returns to the starting vertex with probability 1, and called transient otherwise. The path of a walk on a graph is simply the set of edges this walk has traversed.

Our main result is that the path of a SRW on any graph is a recurrent graph. The proof uses the electrical network interpretation of random walks.

We will give a sketch of the proof, including the necessary background, and discuss related questions, conjectures and results.

Joint work with Itai Benjamini, Russell Lyons and Oded Schramm.

Time:Monday, September 29, 2008 at 2:30 pm.

Location:MEB 243

Speaker:James Lee (University of Washington)

Title:THE SPECTRAL GEOMETRY OF RANDOM GRAPHS

Abstract:Much attention has been paid to the distribution of the eigenvalues of (the adjacency matrix) of random graphs and random discrete matrices, but the eigenvectors of such objects have received somewhat less coverage. We consider the natural question: Do the (normalized) eigenvectors of $G(n,1/2)$ look roughly like random vectors on the sphere? (I don't know, but I'll try to say something non-trivial.)

I will also address some other models, mostly via open questions and irresponsible conjecture, e.g., quantum unique ergodicity for random regular graphs, and so on.

(Joint work with Yael Dekel and Nati Linial.)

Time:Monday, June 2, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Peter Hoff (University of Washington)

Title:Probability models over the Stiefel manifold, with applications to multivariate analysis

Abstract:Normal vectors and orthonormal matrices play an important role in spatial statistics, multivariate analysis and matrix decomposition methods. Probability models for these objects can play a role in describing heterogeneity and sampling variability across multivariate and matrix-valued data. In this talk I describe some simple exponential family models for the set of orthonormal matrices, and illustrate their use in a few data analysis examples. In particular, I will show how to model the heterogeneity in a population of covariance matrices via a model for the heterogeneity in their eigenvector matrices. Estimating the parameters in this model allows us to share information across the different covariance matrices, resulting for improved estimation for each specific covariance matrix.

Time:Monday, May 19, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Oded Schramm (Microsoft Research)

Title:Random metrics, the mathematics of quantum gravity `

Abstract:In quantum gravity, the geometry of space itself is random. One procedure to produce a random metric on a sphere is to fix some large $n$ and to take the uniform distribution over all isometry classes of topological spheres that can be obtained by pasting together side to side $n$ equilateral triangles of side-length $1$. These objects can be shown to be metrically $4$ dimensional in the large (though topologically they are $2$ dimensional). One mathematical reason for the interest in these random geometries stems from the KPZ formula, which is a prediction from physics linking random processes on random geometries to conformally invariant random processes in the plane. Amazingly, some questions turn out to be simpler in the random geometry setting, and the KPZ formula has been used to make very specific conjectures about random processes in the plane. Very recently, there has been a great deal of progress in the mathematical understanding of the random geometries. I plan to survey the subject.

Time:Monday, May 12, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Peter Moerters (University of Bath)

Title:INTERSECTIONS OF RANDOM WALKS IN SUPERCRITICAL DIMENSIONS

Abstract:In high dimensions two independent simple random walks have only a finite number of intersections. In the talk I present recent progress on the problem of describing the exact upper tail behavior of random variables associated with the number of intersections.

Joint work with Xia Chen.

Time:Monday, May 5, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Eyal Lubetzky (Microsoft Research)

Title:Glauber dynamics for the mean-field Ising model and cutoff in birth-and-death chains

Abstract:We study the Curie-Weiss model, i.e., Glauber dynamics for the Ising model on the complete graph on $n$ vertices. It is well-known that the mixing-time in the high temperature regime has order $n\log n$ [Aizenman and Holley (1984)], whereas the mixing-time in low temperatures is exponential in $n$ [Griffiths et al. (1966)]. Recently, Levin, Luczak and Peres proved that for any fixed high temperature there is cutoff, whereas the mixing-time at the critical temperature $\beta=1$ has order $n^{3/2}$. It is natural to ask how the mixing-time transitions from order $n\log n$ (with cutoff) to order $n^{3/2}$ and finally to $\exp\left(\Theta(n)\right)$.

In this work, we extend the results of Levin et al. into a complete characterization of the mixing-time of the dynamics as a continuous function of the temperature, as it approaches its critical point both from below and from above. The above transition between the three regimes appears in the form of a scaling window of order $1/\sqrt{n}$ around the critical temperature. Cutoff occurs only in the subcritical regime, where we determine the cutoff point and window. Furthermore, for each temperature we also determine the order of the spectral gap of the Glauber dynamics.

A key element in the proofs is the analysis of the magnetization chain (the sum of all spins), as it turns out that the mixing of this birth-and-death chain essentially dictates the mixing of entire dynamics. If time permits, I will also present a related general result on the criterion for total-variation cutoff in birth-and-death chains (analogous to the result on convergence in separation by [Diaconis and Saloff-Coste (2006)]).

This is joint work with J. Ding and Y. Peres.

Time:Monday, April 28, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Yuval Peres (Microsoft Research)

Title:Compression exponents of amenable groups and escape rates of random walks

Abstract:Let G be a countable amenable group, equipped with a finite set of generators and a corresponding word metric $d(x,y)$. The Hilbert-space compression exponent $a_*$ of G is the supremum of all $a>0$ such that there exists a Lipschitz map $f$ from G to Hilbert space and a constant $c>0$ so that for all $x,y$ in G, the norm $||f(x)-f(y)||$ is at least $cd(x,y)^a.$ Let $b$ denote the escape rate exponent for simple random walk (X_t) on the Cayley graph of G, that is, suppose that $Ed(X_0,X_t)$ grows like $t^b$. We prove that $2ba_*$ is at most $1$, and equality holds in many cases. In particular for the lamplighter group on Z with integer-valued lamps we have a_*=2/3 and b=3/4. Our proof uses K. Ball's notion of Markov type. We also obtain extensions to $L^p$ compression via p-stable random walks. No prior knowledge of group theory will be assumed; all the relevant notions will be explained in the talk. (Joint work with Assaf Naor and Tim Austin).

Time:Monday, April 21, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Chris Hoffman (University of Washington)

Title:DLA on a cylinder

Abstract:One probabilistic model that has proven notoriously difficult to analyze is diffusion-limited aggregation (DLA) in the plane. This model for the growth of a collection of particles in the plane develops as follows. Initially there is one particle at the origin. Inductively we add particles to the boundary. We start a new particle at infinity and have it perform simple random walk until it is adjacent to the collection. The that particle is added to the collection where it hit the boundary. In this talk we consider a closely related model on a cylinder which we hope will shed light on DLA in the plane. This is joint work with Itai Benjamini.

Time:Monday, April 14, 2008 at 2:30 pm.

Location:LOW 101

Speaker:Ping He (Shanghai University of Finance and Economics and UW)

Title:Excursions of reflecting Brownian motions on Lipschitz domains

Abstract:Since B. Maisonneuve (1975) established an exit system to characterize excursions in general framework, there has been a considerable body of work dealing with what is called the general theory of excursions of a Markov process. However such exit system does not possess analytical description in general. It needs for some application to give an explicit form. P. Hsu (1986) studied the excursions of reflecting Brownian motion on a bounded domain in Euclidean space with smooth boundary. The approach of his work was completely independent of the general theory. Applying to the smoothness of the boundary, the normal derivative for the boundary points which is the density of the Poisson measure with respect to the surface measure is well-defined.

This work is concerned with whether Hsu's elegant results on excursions from the smooth boundary can be extended to the Lipschitz boundary which is no longer smooth. For reflecting Brownian motion on the closure of a bounded Lipschitz domain at first, we take a Martin kernel as Poisson kernel with respect to a positive Radon measure on the boundary. We then investigate the excursions from the boundary straddling on a fixed time $t$ and give a beautiful application on the Feller measure. Based on the previous results, we give an explicit Levy system of the boundary process for the reflecting Brownian motion on Lipschitz domain.

Time:2:30 p.m., Monday, April 7, 2008

Location:LOW 101

Speaker:Zhen-Qing Chen (University of Washington)

Title:A strong limit theorem for Dawson-Watanabe Superprocesses

Abstract:In this talk I will discuss some recent progress on scaling limit theorems for Dawson-Watanabe superprocesses. We will show that weak limit theorem can be established for a large class of superprocesses when the underlying spatial processes are symmetric Hunt processes. When the underling process is a symmetric diffusion with $C^1_b$-coefficients or a symmetric L\'evy process on $\R^d$ whose L\'evy exponent $\Psi (\eta )$ is bounded from below by $c |\eta|^\alpha$ for some $c>0$ and $\alpha \in (0, 2)$ when $|\eta|$ is large, a stronger almost sure limit theorem can be established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrodinger operator. The limit theorems are established under the assumption that an associated Schrodinger operator has a spectral gap.

Joint work with Yanxia Ren and Hao Wang.

Time:2:30 p.m., Monday, March 10, 2008

Location:MEB 245

Speaker:Alexander E. Holroyd (University of British Columbia and Microsoft Research)

Title:SLOW CONVERGENCE IN BOOTSTRAP PERCOLATION

Abstract:Bootstrap percolation is a simple cellular automaton model for nucleation and growth. Sites in an $L$ by $L$ square are initially infected with probability $p$, and a healthy site becomes infected if it has at least 2 infected neighbors. Asymptotically for large $L$, the model is known to undergo a phase transition as the parameter $p \log L$ crosses the threshold $\pi^2/18$. However, simulation predictions for this threshold are typically smaller by more than a factor of two! I'll talk about some attempts to understand this discrepancy. The main result is that the asymptotic value differs from the actual threshold by at least $const/\sqrt{\log L}$. [In contrast, the critical window has width only $const/\log L$.] For a variant model we can prove that to get within 1\% of the asymptotic value, $L$ must be at least $10^{3000}$!

Joint work with Janko Gravner.

Time:2:30 p.m., Monday, Monday, March 3, 2008

Location:MEB 245

Speaker:Ariel Yadin (Weizmann Institute of Science)

Title:THE MAXIMAL PROBABILITY THAT $k$-WISE INDEPENDENT BITS ARE ALL 1

Abstract:A distribution over n bits is called k-wise independent if any subset of k bits are independent. Such distributions arise in many areas in probability and computer science. For example, there is a fundamental connection with linear error-correcting codes.

We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? We use a new approach that does not use coding theory, and thus we are able to improve on previous lower bounds. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros.

Joint work with Ron Peled and Amir Yehudayoff.

Time:2:30 p.m., Friday, February 29, 2008

Location:LOW 102

Speaker:Tadeusz Kuilczycki (Wroclaw University of Technology)

Title:HOT SPOTS THEOREM FOR 2-DIMENSIONAL SLOSHING PROBLEM

Abstract:The sloshing problem is a linear eigenvalue problem that describes the small oscillations of the free surface of a fluid subject to gravity. The sloshing problem is a classical problem in fluid mechanics.

The 2 dimensional sloshing problem has the following physical interpretation. An infinitely long canal with uniform cross section is filled with inviscid heavy fluid (water). The fluid makes, under gravity, oscillations about the equilibrium position. We consider only 2 dimensional motion in planes normal to the generators of the canal bottom.

Let rectangular Cartesian coordinates $(x,y)$ be taken in the plane of the motion with the origin and the $x$-axis in the mean free surface, whereas the $y$-axis is directed upwards. The uniform cross section of the canal is a bounded simple connected domain $W \subset {\bf R} \times (-\infty,0)$, $\partial W = F \cup B$, where $F$, lying on the $x$-axis, is the free surface of the fluid and $B = \partial W \setminus F$, lying in the half plane $y < 0$, is the bottom of the canal. The velocity potential equals $\cos(\sigma_n t + c) u_n(x,y)$. $u_n$ satisfies the following boundary eigenvalue problem $$ \Delta u_n(x,y) = 0, \quad (x,y) \in W, $$ $$ {\partial u_n \over \partial y}(x,0) = - \nu_n u_n(x,0), \quad ( x,0) \in F, $$ $$ {\partial u_n \over \partial n}(x,y) = 0, \quad (x,y) \in B, $$ We have $\sigma_n^2 = \lambda_n g$, where $g$ is the gravitational acceleration. ${\partial\over \partial n}$ is the differentiation normal to $B$.

It is known that there exists a sequence of eigenvalues $$ 0 = \nu_0 < \nu_1 \le \nu_2 \ldots \to \infty. $$

The main result is the following theorem. {\bf Theorem} Let $W \subset F \times (-\infty,0)$ be a bounded convex domain and let $B$ be a smooth curve. Then the first nontrivial eigenfunction $u_1$ is (up to a sign) increasing on $F$, that is $x \to u_1(x,0)$ is increasing for $x \in F$. In particular $u_1$ achieves its minimum and maximum on the boundary of $F$.

The sloshing problem has also the probabilistic interpretation for the semigroup of the Cauchy-like process which is the trace on $F$ of the reflected Brownian motion in $W$.

Time:2:30 p.m., Monday, February 25, 2008

Location:MEB 245

Speaker:Itai Benjamini (Weizmann Institute of Science)

Title:GEOMETRIC REINFORCEMENT

Abstract:We will discuss a few random processes with reinforcement, e.g., DLA.

Time:2:30 p.m., Monday, February 11, 2008

Location:MEB 245

Speaker:Benjamin Weiss (Hebrew University of Jerusalem)

Title:FINITE OBSERVABILITY AND CLASSIFICATION OF ERGODIC PROCESSES

Abstract:A function $F$ defined on some class of stochastic processes $\cal C$ is finitely observable if there is some universal estimation scheme which when applied to longer and longer finite sequences of typical observations drawn from any process $X$ in the class will converge to $F(X)$. In ergodic theory one tries to classify processes up to isomorphism or finitary isomorphism. It turns out that for the class of all ergodic processes there is basically only finitely observable function which is also an isomorphism invariant. All of theses notions will be explained together with some further results and open questions.

Time:2:30 p.m., Wednesday, February 6, 2008

Location:LOW 102

Speaker:Kaneharu Tsuchida (Tohoku University, Japan)

Title:LARGE DEVIATIONS FOR DISCONTINUOUS ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESSES

Abstract:As a useful approach in proving the large deviation principle, the Gartner-Ellis theorem is well known. In this talk, we show the large deviation principle for discontinuous additive functionals of symmetric stable processes by applying this theorem. Joint work with Masayoshi Takeda.

Time:2:30 p.m., Monday, February 4, 2008

Location:MEB 245

Speaker:Maxim Krikun (Institut Elie Cartan, Nancy)

Title:STATIONARY MEASURES FOR BALLISTIC ANNIHILATION WITH BRANCHING

Abstract:A basic model of ballistic annihilation consists of a system of particles in one-dimensional space, moving in either direction with constant speed and annihilating upon collision, so that eventually most of the particles disappear.

When branching is introduced into the picture, the system becomes stable and admits a non-trivial stationary measure. Also, the union of trajectories of all particles, seen as a subset of the plane, has some interesting properties.

This is a joint work with Serguei Popov (USP) and Philippe Chassaing and Lucas Gerin (IECN).

Please note the unusual time and location

Time:2:30 p.m., Wednesday, January 23, 2008

Location:LOW 102

Speaker:Panki Kim (Seoul National University)

Title:BOUNDARY HARNACK PRINCIPLE FOR SUBORDINATE BROWNIAN MOTIONS

Abstract:The boundary Harnack principle for nonnegative classical harmonic functions is a very deep result in potential theory and has very important applications in probability and potential theory. In this talk, we discuss the boundary Harnack inequality for a large class of pure jump symmetric Levy processes, including mixtures of symmetric stable processes.

Time:2:30 p.m., Monday, January 14, 2008

Location:MEB 245

Speaker:Gabor Pete (Microsoft Research)

Title:THE SCALING LIMITS OF DYNAMICAL AND NEAR-CRITICAL PERCOLATION AND THE MINIMAL SPANNING TREE

Abstract:Let each site of the triangular lattice, with small mesh $\eta$, have an independent Poisson clock with rate $\eta^{3/4+o(1)}$ switching between open and closed. Then, at any given moment, the configuration is just critical percolation; in particular, the probability of a left-right open crossing in the unit square is close to 1/2. Furthermore, because of the scaling, the expected number of switches in unit time between having a crossing or not is of unit order.

In joint work with Christophe Garban and Oded Schramm, we prove that the limit (as $\eta \to 0$) of the above process exists as a Markov process, and it is conformally covariant: if we change the domain with a conformal map $\phi(z)$, then time has to be scaled locally by $|\phi'(z)|^{3/4}$. The same proof yields a similar result for near-critical percolation, and it also shows that the scaling limit of (a version of) the Minimal Spanning Tree exists, it is invariant under rotations and scaling, but not under general conformal maps.

Time:2:30 p.m., Monday, January 7, 2008

Location:MEB 245

Speaker:Asaf Nachmias (University of California, Berkeley)

Title:CRITICAL PERCOLATION ON FINITE GRAPH

Abstract:Bond percolation on a graph $G$ with parameter $p$ in [0,1] is the random subgraph $G_p$ of $G$ obtained by independently deleting each edge with probability $1-p$ and retaining it with probability $p$. For many graphs, the size of the largest component of $G_p$ exhibits a phase transition: it changes sharply from logarithmic to linear as p increases. When $G$ is the complete graph, this model is known as the Erdos-Renyi random graph: at the phase transition, i.e. $p=1/n$, the largest component satisfies a power-law of order $2/3$.

For which $d$-regular graphs does percolation with $p=1/(d-1)$ exhibit similar ``mean-field'' behavior? We show that this occurs for graphs where the probability of a non-backtracking random walk to return to its initial location behaves as it does on the complete graph. In particular, the celebrated Lubotzky-Phillips-Sarnak expander graphs and Cartesian products of 2 or 3 complete graphs exhibit mean-field behavior at $p=1/(d-1)$; surprisingly, a product of 4 complete graphs does not.

Please note the unusual time and location

Time:Wednesday, November 28, 2007 at 2:30 pm

Location:MOR 234

Speaker:Toshihiro Uemura (University of Connecticut and University of Hyogo, Japan)

Title:A remark on nonlocal operator

Abstract:Given a Levy kernel, then we reveal a connection between the nonlocal operator having it as Levy measure and the (symmetric) Dirichlet form having it as jumping measure. The relation is obtained through the carre du champ operator associated with the operator. We will show that this relation is quite different from the case of elliptic differential operators (diffusion processes)

Time:Monday November 19, 2007 at 2:30 pm

Location:THO 334

Speaker:Eyal Lubetzky (Microsoft Research)

Title:Poisson approximation for non-backtracking random walks

Abstract:Random walks have been applied extensively in theoretical Computer Science, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. In some of these applications, it seems to be better to apply non-backtracking random walks, rather than simple ones.

In the first part of this talk, I will analyze the mixing rate of non-backtracking random walks on a regular graph. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may indeed be up to twice as fast as the mixing rate of the simple random walk.

The second part of the talk will focus on an application illustrating the fact that non-backtracking walks exhibit stronger pseudo-random properties. We provide a Poisson approximation to the visits to vertices in non-backtracking random walks on expanders, and show that the multi-set of visited vertices is analogous to the result of throwing $n$ balls to $n$ bins uniformly, in contrast to the simple random walk on $G$, which almost surely visits some vertex $\Omega(\log n)$ times.

Based on joint works with Noga Alon, Itai Benjamini and Sasha Sodin.

Time:Friday November 2, 2007 at 2:30 pm

Location:EEB 026

Speaker:Ioana Dumitriu (University of Washington)

Title:Playing golf with many balls: Markov chains and Gittins indices

Abstract:I will analyze and solve a game in which a player chooses which of several Markov chains to advance, with the object of minimizing the expected cost (time) for one of the chains to reach a target state. The solution entails computing a variant of a Gittins index on the states of the individual chains, the minimization of which produces an optimal strategy.

This is older work with Prasad Tetali and Peter Winkler.

Time:Monday October 29, 2007 at 2:30 pm

Location:THO 334

Speaker:Krzysztof Burdzy (University of Washington)

Title:Branching Brownian motion, excursion theory, and potential analysis

Abstract:I will present an open question about a Fleming-Viot-type branching model for Brownian motion. Some progress has been made using excursion theory. Another technical aspect of the partial solution is based on a new type of the boundary Harnack principle.

Joint work with Mariusz Bieniek.

Time:Monday October 15, 2007 at 2:30 pm

Location:LOW 219

Speaker:Zhen-Qing Chen (University of Washington)

Title:Time Reversal and Elliptic Boundary Value Problems

Abstract:In this talk, we will present a new and probabilistic way to solve Dirichlet boundary value problem on bounded Euclidean domains for a class of second order elliptic operators that contain the $div (cu)$ term (dual to the first order term $c\nabla u$). The novelty of our approach is the use of time-reversal of a Girsanov transform. We will derive probabilistic representation for solutions to the elliptic boundary problem. Such a representation also enable us to obtain some new results in PDE.

Joint work with Tusheng Zhang.

Time:Wednesday, May 30, 2007 at 2:30 pm.

Location:MOR 225

Speaker:Mariusz Bieniek (Maria Curie-Sklodowska University, Lublin, Poland)

Title:OPTIMAL BOUNDS FOR EXPECTATIONS OF FUNCTIONS OF GENERALIZED ORDER STATISTICS

Abstract:There is a vast literature devoted to determination of optimal bounds on expectations of order statistics and record values. Most of them are derived by the application of the projection method introduced by Gajek and Rychlik. In this talk we consider extension of these results to generalized order statistics (GOS).

First, we briefly recall the concept of GOS which serves as a unified approach to distribution properties of order statistics, record values and other models of ordered random variables. Then we give a short description of the projection method which relies on finding the projection of functions onto convex cones in the Hilbert space $L^2$ of square integrable functions on $[0,1]$. To find a projection of a function onto the convex cone of nondecreasing functions in $L^2$ we need two tools: the greatest convex minorants method of Moriguti and variation diminishing property of densities of uniform GOS proved by the author. As examples of application of these results we derive optimal bounds on expectations of single GOS and of their differences.

Time:Monday, May 21, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Oded Schramm (Microsoft Research)

Title:THE FOURIER SPECTRUM OF PERCOLATION

Abstract:The indicator function for the existence of a percolation crossing in an n by n square is naturally a function on the discrete cube $\{-1,1\}^E$, where $E$ is the set of edges. As such, it admits a ``Fourier'' expansion. In joint work with Christophe Garban and Gabor Pete, we obtain rather sharp estimates for the ``weight'' of the Fourier coefficients at different frequencies. This allows us to answer some basic questions about dynamical percolation (the dimension of the set of exceptional times and the existence of times with an infinite interface) and about the effect of noise on the crossing event (e.g., if you re-sample the vertical edges, is the event of a crossing in the new configuration substantially correlated with having a crossing in the original configuration?).

The above description is rather vague, and one primary objective of the talk is to clarify and expand it.

Time:Friday, May 18, 2007 at 2:30 pm.

Location:PDL C401

Speaker:Yves Lacroix (Institute of Engineering Sciences of Toulon and The Var)

Title:THE LAW OF SERIES IN ERGODIC THEORY

Abstract:The simplest model of a dynamical system in the probabilistic setting is the one obtained by a stationary ergodic process. For such process, Kolmogorov introduced the notion of entropy, which connects to the one discussed by Shannon when he was working at the Bell laboratories. The entropy is just a positive real number, and it is non zero if and only if the system is non invertible, that is the present does not only depend on the past. In other words, the system is non deterministic.

Any system can be pictured in the probabilistic sense by simple procedures, which are connected to quantification of recurrence to some events. Asymptotically, many processes admit a behavior which is much like the one of the independent process. So I have tried to understand possible asymptotics, in order to go beyond the independent or independent-like cases. After obtaining some characterizations, with T. Downarowicz, we have succeeded in proving that non determinism has an unexpected consequence : asymptotics can only reveal clustering of rare events, which means that positivity of this simple object, entropy, implies that no matter what one tries, rare events will have a natural tendency to cluster... like in the popular but yet unmodelled ``law of series''.

We have even proved that a typical partition of a positive entropy system produces a process with extreme clustering on an upper density one sequence of cylinder lengths. I will try to make this story comprehensive during my talk.

Time:Monday, May 14, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Sourav Chatterjee (University of California, Berkeley)

Title:SPIN GLASSES AND STEIN'S METHOD

Abstract:The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.

Statistics Seminar

Time:Monday, May 7, 2007 at 3:30 pm.

Location:Savery 249

Speaker:Peter Imkeller (Humboldt University at Berlin, Germany)

Title:Scheduling, Percolation, and the Worm Order

Abstract:A spectral analysis of the time series representing average temperatures during the last ice age featuring the Dansgaard-Oeschger events reveals an alpha-stable noise component with an alpha ~ 1.78. Based on this observation, papers in the physics literature attempted a qualitative interpretation by studying diffusion equations that describe simple dynamical systems perturbed by small Levy noise. We study exit and transition problems for solutions of stochastic differential equations and stochastic reaction-diffusion equations derived from this proto-type. Due to the heavy-tail nature of the alpha-stable component of the noise, the results differ strongly from the well known case of purely Gaussian perturbations. For SPDE, transitions are governed by the modes with the largest jumps.

Time:Monday, May 7, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Ron Peled (Microsoft Research)

Title:$k$-WISE INDEPENDENT EVENTS, BOOLEAN FUNCTIONS AND PERCOLATION

Abstract:What is theeffectof $k$-wise independence? We consider some specific Boolean functions whose behavior we understand well when their input is independent random bits (possibly biased). We check whether assuming only that each {\it subset} of size $k$ of their input bits is completely independent can change the distribution of the output of the function significantly. Our main example is percolation: a (bond) percolation on $Z^d$ (the integer lattice in $d$-dimensions), is the process of erasing each edge of $Z^d$ with probability $1-p$ independently of all other edges. It is well known that for each $d\geq 2$, there is a critical $0p_c$, an infinite connected component remains with probability 1 and if $p

We also consider some other examples of boolean functions such as tribes and recursive majority. Our methods use techniques from error correcting codes and the classical moment problem which turn out to have many connections to this subject. We also explore connections to the harmonic analysis of boolean functions. As one application we obtain a new lower bound for the minimal size of an orthogonal array over $GF(q)$.

Time:Monday, April 30, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Yuval Peres (Microsoft Research and University of California, Berkeley)

Title:STRONG SPHERICAL ASYMPTOTICS FOR ROTOR-ROUTER AGGREGATION AND THE DIVISIBLE SANDPILE

Abstract:The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a strong sense. In 2D, for the shape consisting of $n=\pi r^2$ sites, we show that the inradius of the set of occupied sites is at least $r-O(\log r)$, while the outradius is at most $r+O(r^a)$ for any $a > 1/2$. For a related model, the divisible sandpile, we give a constant upper bound for the difference of the outradius and the inradius. For the classical Abelian sandpile model in two dimensions, with $n=\pi r^2$ particles, we improve on the best available bounds due to Le Borgne and Rossin. Similar bounds apply in higher dimensions. (Talk based on joint work with Lionel Levine).

Time:Friday, April 27, 2007 at 2:30 pm.

Location:PDL C401

Speaker:Tomoyuki Shirai (Kyushu University)

Title:ON THE VARIANCE OF RANDOMIZED VALUES OF RIEMANN'S ZETA FUNCTION ON THE CRITICAL LINE

Abstract:Recently, M. Lifshits and M. Weber studied randomized values of Riemann's zeta-function on the critical line $s=1/2 + it$ by sampling along the Cauchy random walk and show that its variance grows slowly, which is related to the famous Lindel\"of hypothesis. In this talk, I would like to explain the background and related topics of this problem, and discuss a slight extension of this result.

PIMS 10th Anniversary Distinguished Lecturer

Time:Thursday, April 19, 2007 at 4 pm.

Location:Smith 304

Speaker:Peter Winkler (Dartmouth College)

Title:Scheduling, Percolation, and the Worm Order

Abstract:When can you schedule a multi-step process without having to take backward steps? Critical are an old concept called "submodularity", a new structure called the "worm order", and a variation of what physicists call "percolation".

With these tools we will attempt to update the computer system at UW, find a lost child in the Cascades, and minimize water usage in Seattle, all without backward steps.

Joint work in part with Graham Brightwell (LSE) and in part with Lizz Moseman (Dartmouth). (This talk is designed to be accessible to undergraduates interested in mathematics.)

Time:Monday, April i16, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Chris Hoffman (University of Washington)

Title:RANDOM WALK AMONG BOUNDED RANDOM CONDUCTANCES

Abstract:Over the last decade there has been much interest in the study of simple random walk on percolation clusters on $Z^d$. Condition on the origin being in the infinite cluster. Then the probability that the simple random walk started at the origin returns to the origin after $2n$ steps is $O(n^{-d/2})$, just as it is for simple random walk on $Z^d$.

In this talk we consider the nearest-neighbor random walk on $Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in (0,1]$. We study the decay of the probability that the random walk started at the origin returns to the origin after $2n$ steps. We prove that this probability is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. We prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. This is joint work with Noam Berger, Marek Biskup and Gady Kozma

Time:Monday, April 9, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Russell Lyons (Indiana University and Microsoft)

Title:SPANNING TREES, RANDOM GRAPHS, AND RANDOM WALKS

Abstract:In the usual Erd\"os-R\'enyi model of random graphs, each pair of $n$ vertices is connected by an edge independently with probability $c/n$ for some constant $c$. When $c > 1$, it has a unique ``giant'' component. How quickly does the number of spanning trees of the giant component grow with $n$ compared to the growth in the number of its vertices? Is it monotonic in $c$? We answer this in joint work with Ron Peled and Oded Schramm.

Time:Monday, April 2, 2007 at 2:30 pm.

Location:MEB 238

Speaker:Jason Swanson (University of Wisconsin, Madison)

Title:STOCHASTIC INTEGRATION WITH RESPECT TO A QUARTIC VARIATION PROCESS

Abstract:Brownian motion (BM) is used to model a wide array of stochastic phenomena in a variety of scientific disciplines. Typically, this is done by using BM as a driving term in a stochastic differential equation (SDE). We are able to define and study these SDEs using Ito's stochastic calculus. Similarly, stochastic partial differential equations (SPDEs) are often used to model stochastic phenomena. In this talk, we consider a very simple example of a stochastic heat equation. The solution to this SPDE, when regarded as a process indexed by time, has a nontrivial 4-variation. It follows that we cannot use the traditional methods of the Ito calculus to define an SDE driven by this process.

In this talk, I will describe work in progress toward constructing a stochastic integral with respect to this process and a corresponding Ito-like change-of-variables formula. The integral being constructed is a limit of discrete Riemann sums. It turns out that the process we are considering has a very close relationship to a certain "flavor" of fractional Brownian motion (FBM). The quest for a calculus for FBM has led researchers in several different directions and there is a large body of literature on the topic. I will discuss some of the similarities and differences between our approach and an analogous approach for FBM using the so-called regularization procedure of Russo and Vallois.

Part of this project is joint work with Chris Burdzy.

Time:Friday March 2, 2007 at 2:30 pm.

Location:PDL C-36

Speaker:Noam Berger (UCLA, visiting Microsoft Research)

Title:Intersection of paths and high dimensional random walk in random environment.

Abstract:We show an easy estimate for the probability of intersection of two random walks in random environment (RWRE) in dimension five or more. We then get two corollaries:a law of large numbers for distributionally symmetric RWRE in dimension geq 5, and a quenched CLT for certain ballistics RWRE-s in dimension geq 4.

Partly based on joint work with Ofer Zeitouni. No knowledge of RWRE will be assumed.

Time:Friday 2/23/2007 at 2:30 pm.

Location:PDL C-401

Speaker:Yan-Xia Ren (Beijing University, visiting University of Oregon)

Title:Limit theorems for super-diffusions corresponding to the operator $Lu+\beta u-ku^2$

Abstract:Consider a superdiffusion $X$ corresponding to the operator $Lu+\beta u-ku^2$, where $\beta(x)$ is bounded from above and is in the Kato class, and $k(x)\ge 0$ is bounded on compact subset of ${\bf R}^d$. Let $-\Lambda $ be the $L^{\infty}$-spectral radius of the semigroup $Q_t$ corresponding to the Schrodinger operator $Lu+\beta u$. We prove that if $\Lambda >0$, the exponential growth rate of the total mass of $X$ is $\Lambda $; if $\Lambda <0$, the exponential decay rate of the total mass of $X$ is $\Lambda <0$. We also describe the limiting behavior of $\exp(-\Lambda t)X_t({\bf R}^d)$, where $X_t({\bf R}^d)$ is the total mass of $X$ at time $t$. In particular, in the case $\Lambda =0$, under some restrict conditions on $\beta$, we give a sufficient and necessary condition for the superdiffusion $X$ exhibiting weak extinction. It turns out that the branching rate function $k$ affects the weak extinction, this should be compared with the known result that $k$ does not affects the weak local extinction of $X$.

Time:Monday 2/12/2007 at 2:30 pm.

Location:SAV 151

Speaker:Alexander Holroyd (University of British Columbia)

Title:Random Sorting Networks

Abstract:See http://www.math.ubc.ca/~holroyd/sort for pictures. Joint work with Omer Angel, Dan Romik and Balint Virag.

Sorting a list of items is perhaps the most celebrated problem in computer science. If one must do this by swapping neighboring pairs, the worst initial condition is when the n items are in reverse order, in which case n choose 2 swaps are needed. A sorting network is any sequence of n choose 2 swaps which achieves this.

We investigate the behavior of a uniformly random n-item sorting network as n->infinity. We prove a law of large numbers for the space-time process of swaps. Exact simulations and heuristic arguments have led us to astonishing conjectures. For example, the half-time permutation matrix appears to be circularly symmetric, while the trajectories of individual items appear to converge to a famous family of smooth curves. We prove the more modest results that, asymptotically, the support of the matrix lies within a certain octagon, while the trajectories are Holder-1/2. A key tool is a connection with Young tableaux.

Time:Monday 2/5/2007 at 2:30 pm.

Location:SAV 151

Speaker:Abraham Flaxman (Microsoft Research)

Title:Expansion and lack thereof in randomly perturbed graphs

Abstract:This talk will investigate the expansion properties of randomly perturbed graphs. These graphs are formed by, for example, adding a random 1-out or very sparse Erdos-Renyi graph to an arbitrary connected graph.

When any connected n-vertex base graph is perturbed by adding a random 1-out then, with high probability, the resulting graph has expansion properties. When the perturbation is by a sparse Erdos-Renyi graph, the expansion of the perturbed graph depends on the structure of the base graph.

The same techniques also apply to bound the expansion in the small worlds graphs described by Watts and Strogatz in [Nature 292 (1998), 440--442] and by Kleinberg in [Proc. of 32nd ACM Symposium on Theory of Computing (2000), 163--170]. Analysis of Kleinberg's model reveals a phase transition: the graph stops being an expander exactly at the point where a decentralized algorithm is effective in constructing a short path.

The proofs of expansion rely on a way of summing over subsets of vertices which allows an argument based on the First Moment Method to succeed.

Time:Monday 1/29/2007 at 2:30 pm.

Location:SAV 151

Speaker:Gabor Pete (Microsoft Research)

Title:Corner percolation on \Z^2, and other linear entropy models

Abstract:Corner percolation is a strongly dependent percolation model introduced by Bálint Tóth, exhibiting some unusual critical behaviour. We prove that all connected components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability \Pr(diameter of the cycle of the origin > n) \approx n^{-\gamma}, and the expectation \E(length of a cycle conditioned on having diameter n) \approx n^\delta, with \gamma=(5-\sqrt{17})/4=0.219... and \delta=(\sqrt{17}+1)/4=1.28... The value of \delta comes from the solution of a singular sixth order ODE, while the relation \gamma+\delta=3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the Additive Brownian Motion, whose level sets have Hausdorff dimension 3/2.

I will discuss many exciting open problems, e.g. on the conformal invariance of some similar linear entropy models.

Time:Monday 1/22/2007 at 2:30 pm.

Location:SAV 151

Speaker:Ioana Dumitriu (University of Washington)

Title:Central Limit Theorems for \beta-Hermite and \beta-Laguerre ensembles (C^1 functions)

Abstract:The \beta-Hermite and -Laguerre ensembles are generalizations of the "classical" Gaussian and central Wishart ones, and have applications in statistical physics (from the theory of log-Coulomb gases to traffic patterns, parked car spacings, etc.)

The (scaled) empirical distributions of these ensembles converge to the famous Wigner semicircle law (in the case of Hermite), respectively, Mar\v{c}enko-Pastur law (for Laguerre). I will show that the fluctuations from these laws describe Gaussian processes on C^1 functions. Then I will comment on a "natural barrier" found at C^{1/2}, and express consternation and a certain degree of frustration with it.

This is joint work with Ofer Zeitouni.

Time:Monday 1/8/2007 at 2:30 pm.

Location:SAV 151

Speaker:Krzysztof Burdzy (University of Washington)

Title:Pathwise uniqueness for reflected Brownian motion in $C^{1+\gamma}$ domains

Abstract:A domain in $R^d$ is called $C^{1+\gamma}$ if its boundary can be locally represented as the graph of a function whose gradient is Holder with exponent $\gamma$. For $d\geq 3$, pathwise uniqueness holds for the SDE representing reflected Brownian motion in a $C^{1+\gamma}$ domain provided $\gamma > 1/2$. For every $\gamma < 1/2$, there exists $d$ and a $C^{1+\gamma}$ domain $D$ in $R^d$ such that pathwise uniqueness fails in $D$.

Work in progress. Joint with R. Bass.

Time:Monday, December 4, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Yuval Peres (Microsoft Research, UC Berkeley and UW)

Title:GRAVITATIONAL ALLOCATION TO POISSON POINTS

Abstract:We consider the Poisson fair allocation problem: Given a realization of a Poisson point process, allocate to each point of the process a unit of volume, in a deterministic translation-invariant way, so that the diameter of the region allocated to each point is stochastically as small as possible. One approach to this problem, studied in previous work with C. Hoffman and A. Holroyd, uses the stable marriage algorithm of Gale and Shapley. Here we show that in dimensions 3 and higher, gravity without inertia yields a very satisfying solution. The argument starts with the classical calculation by Chandrasekar of the total force acting on a point, which has a symmetric stable law. The fairness of the allocation is a consequence of the divergence theorem; The diameters of the allocated regions are analyzed using methods from percolation theory. (Joint work with S. Chatterjee, R. Peled, D. Romik).

Time:Monday, November 27, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Gunnar Gunnarsson (University of Washington)

Title:SPDE MODELS FOR HIGHWAY TRAFFIC

Abstract:We introduce a new stochastic partial differential equation model for multi-lane highway traffic. We prove well-posedness of the model, derive a numerical method for calculating its solutions and estimate some of its parameters.

Time:Monday, November 20, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Yuan-Chung Sheu (National Chiao Tung University, Hsinchu, Taiwan)

Title:AN ODE APPROACH FOR THE EXPECTED DISCOUNTED PENALTY AT RUIN IN JUMP DIFFUSION MODEL

Abstract:For a general penalty function, the expected discounted penalty at ruin was considered by, for example, Gerber and Shiu(1998) and Gerber and Landry (1998) in insurance literature. On the other hand, many pricing functionals in mathematical finance can be formulated in terms of expected discounted penalties. Under the assumption that the asset value follows a jump diffusion, we show the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward phase-type jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general L\'{e}vy measure), we obtain closed form solutions for the expected discounted penalty. As an application, we work out an example in Leland's structural model with jumps. For earlier and related results, see Gerber and Landry(1998), Hilberink and Rogers(2002), Mordecki(2002), Kou and Wang(2004), Asmussen et al.(2004) and others.

Time:Monday, November 13, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Tomoyuki Shirai (Kyushu University)

Title:ON THE NUMBER OF EIGENVALUES OF GINIBRE MATRIX ENSEMBLE

Abstract:Ginibre matrix ensemble is the random matrix whose entries are i.i.d. complex Gaussian random variables. Its random eigenvalues form a determinantal point process (associated with exponential kernel) on the whole complex plane. We discuss the large deviation principle for the number of its eigenvalues inside a ball and also its conditional expectation given that there are no eigenvalues in a big ball.

Time:Monday, November 6, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Tusheng Zhang (University of Manchester)

Title:GLOBAL FLOWS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITHOUT GLOBAL LIPSCHITZ CONDITIONS

Abstract:We consider stochastic differential equations driven by Brownian motion. The coefficients are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift, valid on balls of radius $R$, are supposed to grow not faster than $\log R$, those of diffusion coefficient not faster than $\sqrt{\log R}$. Under these conditions, we establish the existence of a global flow for the stochastic differential equation.

Time:Monday, October 30, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Kavita Ramanan (Carnegie Mellon University)

Title:EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS OF STOCHASTIC DIFFERENTIAL INCLUSIONS

Abstract:We establish sufficient conditions for pathwise uniqueness and existence of strong solutions to a class of stochastic differential equations in $R^n$ with discontinuous drift (interpreted as stochastic differential inclusions), that are possibly reflected in a polyhedral domain with piecewise constant reflection field. In contrast to previous results, we do not impose any uniform ellipticity assumptions, and our results also yield new conditions for uniqueness of solutions to ordinary differential inclusions with drifts that have multiple intersecting surfaces of discontinuity. We will also briefly discuss the application of our results to the study of large deviations of a class of jump Markov processes that arise naturally in applications. This is joint work with Rami Atar and Amarjit Budhiraja.

MATHEMATICS COLLOQUIUM (PIMS 10th Anniversary Distinguished Lecturer)

Time:Tuesday, October 24, 2006 at 4 pm.

Location:Smith 304

Speaker:Gregory Lawler (University of Chicago)

Title:CONFORMAL INVARIANCE AND TWO-DIMENSIONAL STATISTICAL PHYSICS

Abstract:A number of lattice models in two-dimensional statistical physics are conjectured to exhibit conformal invariance in the scaling limit at criticality. In this talk, I will try to explain what the previous sentence means, focusing on three elementary examples: simple random walk, self-avoiding walk, loop-erased random walk. I will describe the limit objects, Schramm-Loewner Evolution (SLE), the Brownian loop soup, and the normalized partition functions, and show how conformal invariance can be used to calculate quantities ("critical exponents") for the model. I will also describe why (in some sense) there is only a one-parameter family of conformally invariant limits. In conformal field theory, this family is parametrized by central charge.

This talk is for a general mathematical audience. No knowledge of statistical physics will be assumed.

Time:Friday, October 20, 2006 at 2:30 pm.

Location:THO 235

Speaker:Eulalia Nualart (Universit\'e Paris 13)

Title:HITTING PROBABILITIES FOR SYSTEMS OF NONLINEAR STOCHASTIC HEAT EQUATIONS

Abstract:In this talk we develop potential theory for a system of non-linear stochastic heat equations in spatial dimension one and driven by a space-time white noise. In particular, we prove upper and lower bounds on hitting probabilities of the process which is solution of this system of equations, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to discuss polarity for points and to compute the Hausdorff dimension of the range and the level sets of this process. In order to prove the hitting probabilities estimates, we need to establish Gaussian type bounds for the bivariate density of the process in order to quantify its degeneracy. For this, we use techniques of Malliavin calculus.

Time:Monday, October 9, 2006 at 2:30 pm.

Location:MGH 288

Speaker:Lionel Levine (University of California, Berkeley)

Title:THE SCALING LIMIT OF DIACONIS-FULTON ADDITION

Abstract:Given two sets $A$ and $B$ in the lattice, the Diaconis-Fulton sum is a random set obtained by putting one particle in every point of the symmetric difference, and two particles in every point of the intersection, of $A$ and $B$. Each "extra" particle performs random walk until it reaches an unoccupied site, where it settles. The law of the resulting random occupied set $A+B$ does not depend on the order of the walks. We find the (deterministic) scaling limit of the sums $A+B$ when $A$ and $B$ consist of the lattice points in some overlapping domains in Euclidean space. The limit is described by focusing on the "odometer" of the process, which solves a free boundary obstacle problem for the Laplacian. Joint work with Yuval Peres.

Time:Monday, October 2, 2006 at 2:30 pm.

Location:MGH 288 (Mary Gates Hall)

Speaker:Peter Moerters (University of Bath)

Title:LOCALIZATION OF MASS IN THE PARABOLIC ANDERSON MODEL

Abstract:The parabolic Anderson model is the Cauchy problem for the heat equation with random potential. After a gentle introduction of some basic features of the model, I will discuss a recent result showing that, for a particular class of potentials, at large times the mass becomes localised in a single point in space, which moves in time.

The talk is based on joint work with W Koenig and N Sidorova.

Time:Thursday 8/3/2006 at 2:30 pm.

Location:LOW 111

Speaker:Panki Kim (University of Illinois at Urbana-Champaign)

Title:Intrinsic Ultracontractivity for Non-symmetric Levy Processes

Abstract:Recently the concept of intrinsic ultracontractivity to non-symmetric semigroups has been introduced by Kim and Song. In this talk, we study the intrinsic ultracontractivity for non-symmetric discontinuous Levy processes.

Time:Wednesday May 24, 2006 at 2:30 pm.

Location:THO 211

Speaker:Takashi Kumagai (Kyoto University)

Title:On the existence of cut points for Brownian motion on fractals

Abstract:Let $B(t)$ be a Brownian motion on some metric space $K$. A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t)\cap B(t,1]$ is empty, and $B(t)$ is called a cut point if $t$ is a cut time. Let $L$ be the set of cut points for $B[0,1]$. When $K=R^d (d=2,3)$, Burdzy showed that non-trivial cut points exist, and later Lawler obtained the Hausdorff dimension of $L$ in terms of the intersection exponent. On the other hand, it is known that when $d=1$, there is no non-trivial cut points. In this talk, we will discuss the existence of cut points for Brownian motion on fractals. We will prove that there is a family of fractals whose Brownian motions are point recurrent, but have non-trivial cut points. This is a on-going joint work with Peter Morters (Bath).

Time:Wednesday May 17, 2006 at 2:30 pm.

Location:THO 211

Speaker:Kavita Ramanan (Carnegie-Mellon University)

Title:A Concentration Inequality for Weakly Contracting Markov Chains

Abstract:Given a Markov chain X on a countable state space S and a Lipschitz function f on S^n, we derive a concentration inequality for the function f around its mean. We use the method of bounded martingale differences to derive this concentration inequality. To set this result in context, we will also provide a brief survey of concentration of measure results in the i.i.d. setting. This is joint work with Leonid Kontorovich.

Time:Monday May 8, 2006 at 2:30 pm.

Location:MEB 238

Speaker:Kittipat Wong (Chulalongkorn University, Thailand)

Title:Large time behavior of Dirichlet heat kernels

Abstract:We will discuss the asymptotic behavior of the transition density $p^D(t,x,y)$ of killed Brownian motions in $D$ where $D \subset R^d, d \ge 2$ is an unbounded domain above the graph of a bounded Lipschitz function.

Time:Wednesday May 3, 2006 at 2:30 pm.

Location:THO 211

Speaker:Rami Atar (Technion and University of Washington)

Title:Large Deviations and Related PDE

Abstract:It is well known that dynamic control problems for Markov processes, in which large deviations are heavily penalized, lead in an appropriate scaling limit to differential games and, in turn, to Hamilton-Jacobi type PDE. I will review results in the field and present new queueing networks applications.

Time:Monday, April 24, 2006 at 2:30 pm.

Location:MEB 238

Speaker:Hanna Jankowski (University of Washington)

Title:Central Limit Theorem for a Zero Range Tagged Particle

Abstract:To further understand the scaling limit of an interacting particle system, one would like to establish a limit law for the evolution of a tagged particle and the associated central limit theorem. Tagging the particles makes this a difficult problem, and one method to deal with it is to consider first an intermediate step: limit laws and fluctuations for the colour empirical densities. I will discuss the nonequilibrium fluctuations of the coloured version of the symmetric zero range particle system, and what this implies about the CLT for the tagged particle.

Time:Monday, April 17, 2006 at 2:30 pm.

Location:MEB 238

Speaker:Krzysztof Burdzy (University of Washington)

Title:Shy couplings

Abstract:Given Markovian transition probabilities, can one construct two processes $X$ and $X'$ with these transition probabilities, such that the distance between $X$ and $X'$ stays always above some $\eps>0$? A pair of proceses with the above property is called a shy coupling. I will give examples of Markovian transition probabilities that admit shy couplings and examples when shy couplings do not exist.

Joint work with I. Benjamini and Z. Chen.

Time:Monday, April 3, 2006 at 2:30 pm.

Location:MEB 238

Speaker:David Wilson (Microsoft Research)

Title:Boundary Connections in Trees and Dimers

Abstract:We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face. Each grove partitions the set of special vertices. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition sigma, Pr[grove has type sigma] / Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise resistances between the special vertices, and Pr[grove has type sigma] / Pr[grove has maximal number of trees] is an integer-coefficient polynomial in the entries of the Dirichlet-to-Neumann matrix. We give analogous polynomial formulas for the double-dimer model. These partition probabilities are relevant to multichordal SLE_8, SLE_2, and SLE_4. (Joint work with Richard Kenyon.)

Note the unusual day of the week.

Time:Friday, March 10, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Jin Ma (Purdue University)

Title:STOCHASTIC CONTROL PROBLEMS DRIVEN BY NORMAL MARTINGALES

Abstract:We study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems, such as the optimal reinsurance selections for general insurance models. The main novel nature of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. Assuming that the driving normal martingale is one-dimensional, we prove the Bellman Principle for such a control problem, and derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. We prove that the value function is the {\it unique} viscosity solution of such an HJB equation and discuss the uniqueness of such PDDE.

This is a join work with Rainer Buckdahn and Catherine Rainer.

Time:Monday, February 27, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Kathryn Temple (Central Washington University)

Title:PARTICLE REPRESENTATIONS OF EXIT MEASURES

Abstract:The connection between diffusions and certain linear elliptic boundary value problems is well-known. More recently, a connection between superdiffusions and a class of nonlinear elliptic PDEs has been developed and exploited. Making this connection requires a construction of the superdiffusion which allows additional structure, such as Dynkin's Branching Exit Markov Systems, the Dawson-Perkins Historical Process, or LeGall's Brownian Snake. We use an adaptation of a particle construction of Kurtz and Rodrigues to construct an exit measure and therefore solutions of the PDEs associated with a certain class of superdiffusions. As time permits, we will give an application of this construction to a simple homogenization problem.

Note the unusual day of the week and unusual room.

Time:Wednesday, February 22, 2006 at 2:30 pm.

Location:SIG 226

Speaker:Yuichi Shiozawa (Tohoku University)

Title:EXTINCTION OF BRANCHING SYMMETRIC $\alpha$-STABLE PROCESSES

Abstract:We give a criterion for extinction of branching symmetric $\alpha$-stable processes in terms of the principal eigenvalue for Schr\"odinger type operators. Here the branching rate and the branching mechanism are spatially dependent. In particular, the branching rate is allowed to be singular with respect to the Lebesgue measure. We also study the exponential growth of the number of particles for these processes.

Time:Monday, February 13, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Oded Schramm (Microsoft)

Title:THE DISCRETE GAUSSIAN FREE FIELD AND ITS LEVEL LINES

Abstract:An instance of the Gaussian free field is a random function defined in a domain in $R^d$. It is a generalization of Brownian motion to the case where time is multi-dimensional, and it is useful for modelling many different kinds of random surfaces. In two dimension, it exhibits conformal invariance.

During the talk we will define the Gaussian free field and also the discrete Gaussian free field and describe joint work with Scott Sheffield identifying the scaling limit of the level lines of the discrete Gaussian free field in two dimensions (as the lattice mesh tends to zero). These limits are random curves having Hausdorff dimension 3/2 a.s.

Time:Monday, February 6, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Nicole Immorlica (Microsoft)

Title:MARRIAGE, HONESTY AND STABILITY

Abstract:Many centralized two-sided markets, such as the medical residency market or online dating services, form a matching between participants by running a stable marriage algorithm. It is a well-known fact that no matching mechanism based on a stable marriage algorithm can guarantee truthfulness as a dominant strategy for participants. However, as we will show in this talk, in a certain probabilistic setting, truthfulness is (in some sense) the best strategy for the participants.

We show this by proving that in our setting the set of stable marriages is small. We derive several corollaries of this result. First, we show that, with high probability, in a stable marriage mechanism, the truthful strategy is the best response for a given player when the other players are truthful. Then we analyze equilibria of the deferred acceptance stable marriage game. We show that the game with complete information has an equilibrium in which a $(1-o(1))$ fraction of the strategies are truthful in expectation. In the more realistic setting of a game of incomplete information, we will show that the set of truthful strategies form a $(1+o(1))$-approximate Bayesian-Nash equilibrium. Our results have implications in many practical settings and were inspired by experimental observations in a paper of Roth and Peranson (1999) concerning the National Residency Matching Program.

This is joint work with Mohammad Mahdian.

Time:Monday, January 30, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Omer Angel (University of British Columbia)

Title:ONE DIMENSIONAL DLA

Abstract:We consider a variation on DLA (diffusion limited aggregation) in 1 dimension generated by a random walk with large jumps. The growth rate of the diameter of the $n$ particle aggregate depends on the tail of the step distribution, and exhibits three phase transitions when the steps have 1, 2 or 3 finite moments.

Time:Monday, January 23,, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Rami Atar (Technion and University of Washington)

Title:ANALYSIS OF MIRROR COUPLINGS IN SMOOTH DOMAINS

Abstract:We analyze a coupling of two reflecting Brownian motions in (the closure of) a smooth domain, with the property that whenever both processes are in the domain, the motions form mirror images of each other. Under appropriate geometric conditions, we show that the motion of the mirror is limited in the sense that there are parts of the domain that it will never intersect (for suitable initial conditions). This will be described along with consequences regarding monotonicity of the Neumann-Laplacian eigenfunctions a la the `hot spots' problem. Joint work with Chris Burdzy.

Note the unusual day of the week and time. This will be also a colloquium talk.

Time:4:00 p.m., Thursday, January 19, 2006

Location:Smith 105

Speaker:Jin Feng (University of Massachusetts at Amherst)

Title:LARGE DEVIATIONS FOR MARKOV PROCESSES AND RELATED VARIATIONAL PROBLEMS

Abstract:Large deviation estimates are probabilistic limit theorems which are used to describe atypical behavior of random systems. Markov processes are a rich class of probabilistic models. Their generators constitute a link between probability, classical analysis and *linear* partial differential equations. I will describe a method for deriving large deviation estimates for a sequence of Markov processes through convergence of some *nonlinear* transforms of their generators. This is a previously unknown connection between probability and certain topics in nonlinear analysis such as Hamilton-Jacobi equations, viscosity solutions, and optimal control theory. I will offer examples where probability can help with analytical problems and vice versa. I will use examples ranging from small random perturbations of ODEs to the more physically motivated ones such as macroscopic description of multi-scale microscopic interacting particle systems.

Time:Monday, January 9, 2006 at 2:30 pm.

Location:PDL C-401

Speaker:Jason Swanson (University of Wisconsin, Madison)

Title:ASYMPTOTIC BEHAVIOR OF A GENERALIZED TCP CONGESTION AVOIDANCE ALGORITHM

Abstract:The Transmission Control Protocol (TCP) is designed to guarantee reliable and sequential delivery of packets of data from a sender to a receiver on a computer network. To avoid network congestion, TCP uses various variations on an additive-increase-multiplicative-decrease algorithm, which means that the rate of data flow is increased linearly until a packet loss is detected, at which point the rate is reduced by some constant multiplicative factor. I will present a generalization of this TCP Congestion Avoidance algorithm, which is due to Teunis J. Ott of the New Jersey Institute of Technology. This model includes, as a special case, the so-called "Scalable TCP," which is a multiplicative-increase-multiplicative-decrease scheme.

The general model is given by a Markov chain $\{W_n\}$, which represents the size of the congestion window (that is, the rate of data flow) at the time of the delivery of the $n$-th packet. Implicit in this model is a parameter $p$, which denotes the probability of a packet loss. Under suitable scaling of time and space, this process converges weakly as $p\to 0$ to the solution of a stochastic differential equation. The form of the equation depends on the values of certain parameters in the model. For some values, it is the Ornstein-Uhlenbeck equation; for the others, it is an equation driven by a Poisson process. I will sketch the proof of this result and discuss further unanswered question. This is joint work with Teun Ott.

Time:Wednesday, December 7, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Jeremy Quastel (University of Toronto)

Title:Asymptotic speed of traveling fronts in the KPP equation with noise

Abstract:The most basic reaction-diffusion equation, KPP, was introduced by Fisher to model the spread of an advantageous gene. In this context it is very natural to consider what happens to traveling fronts when the equation is perturbed by an appropriate noise. Brunet and Derrida observed, by simulations and physical arguments, that the noise leads to an unexpectedly large slowdown of the traveling fronts. We describe the problem and proofs of some of their conjectures. This is joint work with Carl Mueller (Rochester) and Leonid Mytnik (Technion).

Time:Monday, November 21, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Jim Burke (University of Washington)

Title:Estimation of Constrained Mixture Densities

Abstract:We consider the constrained mixture density estimation problem over a parametric family of densities. It is shown how this problem can be posed as a convex optimization problem on the space of regular Borel measures. By combining this formulation with Caratheodory's Theorem for convex sets in finite dimensions it is possible to pose an equivalent finite dimensional optimization problem. The problem is then amenable to a variety of numerical optimization routines. One such approach based on interior point technology is discussed and numerical results are presented.

This research is the result of a joint work between the speaker and Yeongcheon Baek.

Time:Monday, November 14, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Peter D. Hoff (University of Washington)

Title:Random Graph Models for Relational Network Data

Abstract:Relational data measure the presence or absence of links among a set of objects. This data structure is very general and has many applications in the social and biological sciences. In this talk we consider two families of probability models which are used to make inference from network data. The first family includes exponentially parameterized models, which are motivated by simplicity and maximum entropy considerations. Models in the second family are motivated by a certain type of row-and-column exchangeability for arrays, and typically have a very large parameter space. Finally, we compare the appropriateness of these two model classes for a variety of inferential goals.

Time:Monday, November 7, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Chris Hoffman (University of Washington)

Title:Dynamic random walk in two dimesnions

Abstract:Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks in d dimensions, S_n(t), indexed by a real number t. They asked which properties of simple random walk that hold for almost every t hold for all t. They proved that if d=3,4 then there exist (random) times t such that the random walk returns to the origin infinitely often, but if d>4 then the random walk returns to the origin only finitely often for all t. We prove that if d=2 then there exists (random) times t such that the random walk returns to the origin only finitely often. This result is related to a theorem of Adelman, Burdzy and Pemantle in which they prove that although Browninan motion in three dimensions spends an infinite amount of time in a given infinite cylinder of radius one almost surely, but there exist (random) cylinders where Brownian motion spends only a finite amount of time almost surely.

Time:Monday, Oct 31, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Vlada Limic (University of British Columbia)

Title:The spatial Lambda-coalescent

Abstract:This talk is based on a joint paper with Anja Sturm, and it will describe the extension of the Lambda-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Lambda-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Lambda-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial Lambda-coalescents on large tori in transient dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context.

Time:Monday, Oct 24, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:David Levin (University of Oregon)

Title:A Coupling, and a Conjecture of Darling and Erdos

Abstract:I will describe a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit iid sequence. This coupling can be used to resolve a conjecture of Darling and Erdos (1956). Joint work with D. Khoshnevisan.

Time:Monday, Oct 17, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Ken Alexander (University of Southern California)

Title:The pinning transition for a polymer in the presence of a random potential

Abstract:We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume the probability of an excursion of length $n$ from 0, in the absence of the potential, decays like $n^{-c}$ for some $c>1$. Disorder is introduced by, having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction, called the contact fraction, of its monomers at 0 with high probability. We obtain bounds for the contact fraction near the critical point and examine the effect of the disorder on the specific heat exponent, which describes the approach to 0 of the contact fraction at the critical point. Our results are consistent with predictions in the physics literature that the effect of disorder is quite different in the cases $c<3/2$ and $c>3/2$.

Time:Wednesday, Oct 12, 2005 at 2:30 pm.

Location:SMI 311

Speaker:Bartek Dyda (Technical Univ of Wroclaw)

Title:On fractional order Hardy inequalities

Abstract:We will state the fractional order Hardy inequality for e.g. bounded Lipschitz domains, and prove it in the simplest case when the domain is R^d \setminus \{0\}. This inequality will give us transcience of the censored stable process (with the stability index > 1) in bounded Lipschitz domains. This result was obtained earlier (by another methods) in the paper of Bogdan, Burdzy and Chen. We will also mention some further results concerning fractional order Hardy inequalities.

Time:Wednesday, August 24, 2005 at 3:30 pm.

Location:Padelford C-36

Speaker:Wojbor A. Woyczy\'nski (Case Western Reserve University)

Title:STOCHASTIC PARTICLE METHODS FOR NONLINEAR EVOLUTION EQUATIONS INVOLVING L'EVY GENERATORS

Abstract:I will consider a class of nonlinear integro-differential equations driven by L\'evy infinitesimal generators and involving nonlocal nonlinearities represented by singular integral operators. We associate a nonlinear diffusion in the McKean sense with the original singular equation and are able to prove the convergence of cut-off interacting particle systems to the law of this nonlinear singular diffusion.

Time:Monday, May 23, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Elton Hsu (Northwestern University)

Title:MASS TRANSPORTATION PROBLEM AND COUPLING OF DIFFUSION PROCESSES

Abstract:We will discuss the problem of existence and uniqueness of maximal Markov coupling of diffusion processes, in particular, Brownian motion on Riemannian manifolds. We will show how this problem is related to the mass transportation problem with a cost function determined by the heat kernel of the diffusion process. The problem is completely solved only in the Euclidean case.A part of this talk is the joint work with Theo Sturm (Bonn).

Time:Monday, May 16, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Matthew Stephens (University of Washington)

Title:EXCHANGEABLE AND NON-EXCHANGEABLE DISTRIBUTIONS ARISING FROM APPLICATIONS IN POPULATION GENETICS

Abstract:Population genetics is the study of genetic variation in random samples of individuals from a population. I will give an informal introduction to some of the mathematical theory underlying recent progress in this discipline. Much of the theory is derived under simplistic and unrealistic assumptions. I will describe work to relax these assumptions to capture important aspects of real data. Although this work has had important impact in several practical applications, it raises several interesting and unresolved theoretical problems.

Time:Monday, May 9, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Karoly Simon (Technical University of Budapest)

Title:THE SIZE OF THE ALGEBRAIC DIFFERENCE OF TWO RANDOM CANTOR SETS

Abstract:In this talk we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one, implies the existence of interior points in the difference set. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a construction of random Cantor sets similar to the Mandelbrot percolation but with different probabilities.

Time:Monday, May 2, 2005 at 4:30 pm.

Location:THO 119

Speaker:Michael Taksar (University of Missouri)

Title:SINGULAR STOCHASTIC CONTROL AND RELATED PDE WITH GRADIENT CONSTRAINTS IN PORTFOLIO OPTIMIZATION MODELS IN MATHEMATICAL FINANCE

Abstract:In the modern mathematical finance the stock prices are modeled by stochastic differential equations, whose solutions produce logarithmic Brownian motions. This is the backbone of what is nowadays became the classical Black-Scholes option pricing theory and Merton's investment/consumption theory. We consider a dynamical portfolio optimization model in the spirit of the latter. The portfolio consists of several risky assets (Stocks) and one risk-free asset (Bond). The rate of return on Bond is constant while the rate of return of Stocks is governed by SDE of the logarithmic Brownian motion type. Funds can be transferred from one asset to another, however such transaction involves penalty (brokerage fees) proportional to the size of the transaction. The objective is to find the policy which maximizes the expected rate of growth of funds.The main mathematical tool in the solution of this problem is singular stochastic control theory. In this theory the control functionals are represented by processes of bounded variation, and the optimal control consists of functionals which reflect the process from an a priori unknown boundary. They are continuous but singular (not absolutely continuous) with respect to time. The analytical part of the solution to singular control is related to a free boundary problem for an elliptic PDE with gradient constraints, similar to the ones encountered in elastic-plastic torsion problems. The existence of the classical $C^2$ solution cannot be proved in general but one can show an existence of viscosity solution to this equation.

The optimal policy is to keep the vector of fractions of funds invested in different assets in an optimal (a priori unknown) boundary. We show how to find these boundaries explicitly in the case of one risky and one risk-free asset when the problem becomes one dimensional. In this case the free boundary problem can be reduced to a Stephan problem for an ODE.

Time:Thursday, April 21, 2005 at 2:30 pm.

Location:SAV 131

Speaker:Wlodzimierz Bryc (University of Cincinnati)

Title:QUADRATIC HARNESSES, Q-COMMUTATIONS, AND ORTHOGONAL MARTINGALE POLYNOMIALS

Abstract:Harnesses were introduced by Hammersley (1967) as random fields that model `long-range misorientation'. They were studied for example by Williams (1973), and recently by Mansuy-Yor (2005) who considered harnesses indexed by t>0 and defined them by a reverse-martingale condition for the normalized increments of the process.Quadratic harnesses are related to Hammersley-Manuy-Yor harnesses by mimicking the relation between the martingale and the quadratic martingale conditions. Examples of quadratic harnesses are the Poisson, Gamma, Pascal (negative binomial), and the Wiener process. Lesser known examples are Markov processes associated with certain free Levy processes, and with non-commutative q-Gaussian processes introduced in the physics literature by Frisch-Bourret (1970) and constructed rigorously by Bozejko-Kummerer-Speicher {1997}.

It is plausible that generic quadratic harnesses are in fact five-parameter Markov processes. The five parameters arise as the coefficients in the q-commutation equation for the Jacobi matrix of the associated orthogonal martingale polynomials, which exist under minimal assumptions. At present, constructions of the corresponding Markov transition probabilities are known only for special choices of these parameters, and explicit answers are even less frequent. One such explicit example of a two-parameter quadratic harness is obtained by appropriately re-scaling a certain non-homogeneous pure-birth Markov process that `explodes' at a deterministic moment of time, and then `returns back from infinity' as a pure-death process.

This talk is based on joint research with W. Matysiak and J. Wesolowski.

Time:Monday, April 18, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Fausto Di Biase (Universit\`a `G.d'Annunzio', Pescara, Italy)

Title:THE STOLZ APPROACH IS SHARP, ISN'T IT?

Abstract:We consider the following question: how sharp is the Stolz approach region for the almost everywhere convergence of bounded harmonic functions in the unit disc in the plane or in NTA domains in $R^n$? The issue was first settled in the rotation invariant case in the unit disc by Littlewood in 1927 and later examined, under less stringent conditions, by Aikawa in 1991. We will present results that are, in a certain sense, sharp.A joint work in collaboration with A. Stokolos, O. Svensson and T. Weiss.

Time:Monday, April 11, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Joan Lind (University of Washington)

Title:THE GEOMETRY OF THE LOEWNER EVOLUTION

Abstract:The Loewner differential equation provides a connection between continuous, real-valued functions (called driving terms) and families of domains in the complex plane (said to be generated by the driving term.) The recent interest in this 80-year-old equation is due to Oded Schramm's introduction of the SLE processes, which have allowed mathematicians to prove many results predicted by physicists as well as Mandelbrot's conjecture that the outer boundary of a planar Brownian path has Hausdorff dimension 4/3. I will discuss how the geometry of the generated domains is related to the driving function, both in the deterministic setting and for SLE.

Time:Monday, April 4, 2005 at 2:30 pm.

Location:SMI 102

Speaker:Julien Dub\'edat (Courant Institute, New York University)

Title:COMMUTATION OF SLE's

Abstract:Schramm-Loewner Evolutions (SLEs) have proved to be a powerful tool to describe the scaling limit of a conformally invariant simple curve. In several instances (percolation, uniform spanning tree ...), one can define in a discrete setting several simple curves. We will discuss questions pertaining to the joint law of these curves in the scaling limit.

Time:Tuesday, March 22, 2005 at 2:30 pm.

Location:PDL C-36

Speaker:Jason Swanson (University of Wisconsin)

Title:WEAK CONVERGENCE OF THE MEDIAN OF INDEPENDENT BROWNIAN MOTIONS

Abstract:In a model of Spitzer (1968) model, we begin with countably many particles distributed on the real line according to a Poisson distribution. Each particle then begins moving with a random velocity; these velocities are independent and have mean zero. The particles interact through elastic collisions: when particles meet, they exchange trajectories. We then choose a particular particle (the tagged particle) and let X(t) denotes its trajectory. Spitzer showed that, under a suitable rescaling of time and space, X(t) converges weakly to Brownian motion.Harris (1965) demonstrated a similar result: if the individual particles move according to Brownian motion, then X(t) converges to fractional Brownian motion. These results were generalized even further by D\"{u}rr, Goldstein, and Lebowitz in 1985.

All of these results rely heavily on the initial distribution of the particles. The initial Poisson distribution provides tractable computations in the models of Spitzer, Harris, and D\"{u}rr et al;

In this talk, I will outline the following result: the (scaled) median of independent Brownian motions converges to a centered Gaussian process whose covariance function can be written down explicitly. As with the other results, proving tightness is the chief difficulty. Unlike the other results, the initial distribution of the particles does not play a key role. Tightness is proved in this model by direct estimates on the median process itself. It is my hope that these techniques can be generalized and used to extend the current family of results to models with arbitrary initial particle distributions.

Time:Monday, March 7, 2005 at 2:30 pm.

Location:PDL C-401

Speaker:Roman Kotecky (Charles University, Prague, and Microsoft Research)

Title:BIRTH OF EQUILIBRIUM DROPLET

Abstract:We consider large deviations for Ising model in the phase coexistence region. The typical behavior, inside the coexistence region and far from its edge, is governed by droplet configurations. However, a new type of transition is experienced close to the edge of the coexistence. It stems from the competition between droplet contributions and supersaturated phase and leads to an abrupt occurrence of a droplet. Its proof needs a careful evaluation of typical configurations in different regimes. Main results will be explained and some idea of the proofs will be given without assuming any preliminary knowledge about the Ising model.The talk is based on joint papers with Marek Biskup and Lincoln Chayes as well as a recent work with Ostap Hryniv and Dima Ioffe.

Time:Monday, February 28, 2005 at 2:30 pm.

Location:THO 135

Speaker:Ryan O'Donnell (Microsoft Research)

Title:NOISE STABILITY OF BOOLEAN FUNCTIONS WITH LOW INFLUENCES

Abstract:Suppose we model a close election (such as Bush/Gore '00 or Gregoire/Rossi '04) by assuming n voters vote independently and 50-50 between two candidates, 0 and 1. An ``election scheme'' is a boolean function $f : \{0,1\}^n \to \{0,1\}$ mapping the votes to a winner; e.g., $f$ = Majority, or $f$ = Electoral College. Analyzing the properties of such functions is hampered by the fact that the trivial ``dictator'' schemes, $f(x_1, ..., x_n) = x_i$, are often extremal. It is thus both natural and desirable to restrict attention to $f$'s in which each voter has small ``influence''.We resolve two conjectures regarding functions with low influences: 1. The ``Majority Is Stablest'' conjecture from theoretical computer science, which states that if each vote is mis-recorded independently with probability epsilon (butterfly ballot? Diebold?) then Majority is the low-influence election scheme most likely to preserve the outcome. 2. The ``It Ain't Over Till It's Over'' conjecture from social choice in economics, which states that if a random 99\% of the votes are in (from Florida, say) then for any low-influence election scheme, the outcome is still overwhelmingly likely to be ``too close to call''.

This is joint work with Elchanan Mossel (Berkeley) and Krzysztof Oleszkiewicz (Warsaw).

Time:Tuesday, February 22, 2005 at 2:30 pm.

Location:SAV 311

Speaker:Dayue Chen (Peking University)

Title:THE ANCHORED EXPANSION CONSTANT OF RANDOM GRAPHS

Abstract:The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virag (2000). We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1. We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.This talk is based on my joint work with Yuval Peres and Gabor Pete of the University of California, Berkeley. The paper entitled ``Anchored Expansion, Percolation and Speed'' just appeared in the Annals of Probability.

Time:Monday, February 14, 2005 at 2:30 pm.

Location:THO 135

Speaker:Assaf Naor (Microsoft Research)

Title:MARKOV CHAINS IN METRIC SPACES AND THEIR APPLICATIONS TO METRIC GEOMETRY

Abstract:In this talk we will discuss the notion of Markov type: an important bi-Lipschitz invariant of metric spaces which was introduced by K. Ball. This invariant measures the affect of the geometry of a metric space $X$ on the speed of certain Markov chains taking values in $X$. We will describe some of the applications of Markov type, and proceed to present a proof of the recent solution of Ball's Markov type 2 problem, showing that $L_p, p>2$, has Markov type 2. This result completes the work of Ball and settles a conjecture of Johnson and Lindestrauss from 1992 by showing that every Lipschitz function defined on a subset of $L_p, p>2$, with values in $L_q, 1 < q <2$, extends to a Lipschitz function defined on all of $L_p$.Based on joint work with Yuval Peres, Oded Schramm and Scott Sheffield.

Time:Monday, February 7, 2005 at 2:30 pm.

Location:THO 135

Speaker:Uriel Feige (Weizmann Institute and Microsoft Research)

Title:ON SUMS OF INDEPENDENT RANDOM VARIABLES

Abstract:We prove the following inequality: for every positive integer $n$ and every collection $X_1, \ldots, X_n$ of nonnegative independent random variables that each has expectation~1, the probability that their sum remains below $n+1$ is at least $\alpha > 0$. Our proof produces a value of $\alpha = 1/13 \simeq 0.077$, but we conjecture that the inequality also holds with $\alpha = 1/e \simeq 0.368$.

Time:Monday, January 31, 2005 at 2:30 pm.

Location:THO 135

Speaker:Richard Bass (University of Connecticut)

Title:RENORMALIZED SELF-INTERSECTION LOCAL TIME AND THE RANGE OF RANDOM WALKS

Abstract:Self-intersection local time $\beta_t$ is a measure of how often a Brownian motion (or other process) crosses itself. Since Brownian motion in the plane intersects itself so often, a renormalization is needed in order to get something finite. LeGall proved that $E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite for large $\gamma$. It turns out that the critical value is related to the best constant in a Gagliardo-Nirenberg inequality. I will discuss this result (joint work with Xia Chen) as well as large deviations for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$. The range of random walks is closely related to self-intersection local times, and I will also discuss joint work with Jay Rosen making this idea precise.

Time:Tuesday, January 25, 2005 at 12:30 pm.

Location:MOR 219

Speaker:Bruce Erickson (University of Washington)

Title:FINITENESS OF INTEGRALS OF FUNCTIONS OF LEVY PROCESSES

Abstract:We determine necessary and sufficient conditions under which various integrals of functions of the supremum absolute value of a Levy process are finite. The conditions are expressed analytically in terms of the canonical Levy measure of the process. An application of some of these results leads to an interesting local (small time) comparison of the above mentioned supremum process with the supremum absolute jump process.This is a brief report on some work in progress with R. Maller.

Time:Wednesday, January 12, 2005 at 3:30 pm.

Location:Padelford C-36

Speaker:Krzysztof Burdzy (University of Washington)

Title:THE ROBIN PROBLEM AND RAY-KNIGHT THEOREM

Abstract:The ``Robin problem'' or the ``third boundary problem'' is a mathematical model for the flow of a substance (or heat) out of a domain through a semi-permeable membrane. I will address the question of when the concentration of the substance (or the temperature) is bounded below by a constant over the whole domain. The argument is based in part on a very non-trivial (although old) result known as "Ray-Knight theorem". It describes the distribution of the local time of Brownian motion as a function of the space variable.

Time:Monday, Dec 6, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:David Wilson (Microsoft Research)

Title:BALANCED BOOLEAN FUNCTIONS THAT CAN BE EVALUATED SO THAT EVERY INPUT BIT IS UNLIKELY TO BE READ

Abstract:A Boolean function of n bits is balanced if it takes the value 1 with probability 1/2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random inputs, no input bit is read with probability more than $\Theta(n^{-1/2} \sqrt{\log n})$. We give a balanced monotone Boolean function for which the corresponding probability is $\Theta(n^{-1/3} \log n)$. We then show that for any randomized algorithm for evaluating a balanced Boolean function, when the input bits are uniformly random, there is some input bit that is read with probability at least $\Theta(n^{-1/2})$. For balanced monotone Boolean functions, there is some input bit that is read with probability at least $\Theta(n^{-1/3})$.Joint work with Itai Benjamini and Oded Schramm.

Time:Monday, Nov. 29, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:David White (University of Washington)

Title:INTERACTIONS OF A BROWNIAN PARTICLE AND AN INERT PARTICLE

Abstract:Frank Knight recently introduced a model for the motion of an inert particle that is impinged on one side by a Brownian particle. Interesting behavior can result when the two types of particles are combined in different configurations. We describe the stochastic process resulting from these interactions for a few possible configurations.

Time:Monday, Nov. 15 and Monday, Nov. 22, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Charles J. Geyer (University of Minnesota)

Title:PROBABILITY AND STATISTICS USING NELSON-STYLE NONSTANDARD ANALYSIS (parts I and II)

Abstract:Edward Nelson's book Radically Elementary Probability Theory shows how to `radically' simplify both nonstandard analysis (NSA) and probability theory. In this approach all sample spaces are finite and all nonempty events have nonzero probability. So all nontrivial collections of random variables are finite. We have only with finite sequences of random variables, stochastic processes with finite carriers, etc. This eliminates any need for measure theory. Nevertheless, infinitesimal and unlimited numbers allow analogs of phenomena of conventional probability theory, such as continuous random variables, laws of large numbers, central limit theorems, invariance principles, and Brownian motion.In part I (Nov 15) we will introduce Nelson's `radically elementary' version of NSA, which is simple enough to teach to undergraduates, and using de Finetti's theorem as an example, show how this approach can greatly simplify probabilistic reasoning. We also start a survey of the definitions and main results of Nelson's book.

In part II (Nov 22) we will discuss some of our work: weak convergence in metric spaces, the portmanteau theorem, the continuous mapping theorem, the Glivenko-Cantelli theorem, Prohorov metric strong consistency of the empirical process, spatial point processes (redone Nelson-style).

Time:Monday, Nov. 8, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Benjamin Morris (Indiana University and Microsoft Research)

Title:THE MIXING TIME FOR THE THORP SHUFFLE

Abstract:In 1973, Thorp introduced the following card shuffling procedure. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip; then drop from the other pile. Continue this way, flipping an independent coin each time, until both piles are empty.Despite its simple description, the Thorp shuffle has been hard to analyze. It has long been believed that the mixing time is polynomial in log of the number of cards. We prove the first such bound.

Time:Monday, Nov. 1, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Chris Hoffman (University of Washington)

Title:HOW WELL CAN WE PREDICT A NEAREST NEIGHBOR PROCESS?

Abstract:We consider the class of integer valued nearest neighbor processes. These are the processes $S(t)$ such that for all $t$ $|S(t)-S(t+1)|=1$ a.s. In this problem we are given the past $S(0),S(-1),...$ and we try to predict $S(k)$. We want to bound the maximum probability that we predict correctly. Define $$v(k)=\sup_{n,S(0),S(-1),S(-2),\dots} P(S(0)-S(k)=n|S(0),S(-1),S(-2),\dots).$$ We will show that for any decreasing sequence $f(k)$ such that $\sum_{k=1}^{\infty}f(k)=\infty$ there does not exist a nearest neighbor process such that $$v(k)<{1\over kf(k)}$$ for all $k$ and that this condition is tight. We will also talk about how this problem relates to random walks on percolation clusters.

Time:Monday, Oct 18, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Zhenqing Chen (University of Washington)

Title:EIGENVALUES FOR SUBORDINATE PROCESSES IN DOMAINS

Abstract:Abstract: When studying spectral properties for nonlocal operator in domains, one often faces many new challenges. For example, in contrast to the Laplace operator case, even the first eigenvalue for 1-dimensional fractional Laplace operator in unit interval is not explicitly known. In this talk, we will show that it is possible to obtain two-sided eigenvalue estimates for fractional Laplacian in terms of the eigenvalues of the Dirichlet Laplacian in the domain. In fact, the above result holds more generally, with Laplacian being replaced by a self-adjoint operator $L$ that is the generator of a strong Markov process, and the fractional Laplacian being replaced by $-\phi (-L)$, where $\phi$ is the Laplace exponent of a subordinator. We further show that the eigenvalues of $\phi (-L)$ in a domain with Dirichlet exterior condition depends continuously on $\phi$. This talk is based on joint work with R. Song.

Time:Monday, Oct 11, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Nati Linial (Hebrew University of Jerusalem and Microsoft)

Title:RANDOM LIFTS OF GRAPHS AND WHAT THEY ARE GOOD FOR

Abstract:

Time:Monday, Oct 11, 2004 at 2:30 pm.

Location:PDL C-401

Speaker:Nati Linial (Hebrew University of Jerusalem and Microsoft)

Title:RANDOM LIFTS OF GRAPHS AND WHAT THEY ARE GOOD FOR

Abstract:The most thoroughly studied random graph model is the classical Erd\"os-Renyi G(n,p) model where edges are placed independently and with probability p between any pair from among n vertices. There is a strong feeling in many parts of discrete mathematics that this is not enough, and new models for random graphs are needed. The quest for new models often comes from a desire to depict some physical phenomena, or just to understand the typical behavior of a certain family of graphs. Random lifts are a new class of random graphs, and the chief reason for introducing them was the hope to construct graphs with special desired properties, which existing methods seem unable to achieve. Our research into these graphs has revealed lots of beautiful phenomena and raised new intriguing problems. However, only recently did it become possible to use Random Lifts to construct graphs "by design" i.e., graphs with some desired extremal properties. Specifically, they were utilized to explicitly construct graphs with very good spectral properties ("Quasi-Ramanujan" graphs). In this talk I will explain what random lifts are. I will also survey the main known results and some of the open problems.

Time:Monday, October 4, 2004 at 2:30 pm.

Location:PDL C-36

Speaker:Krzysztof Burdzy (University of Washington)

Title:LENSES IN SKEW BROWNIAN MOTION`

Abstract:Skew Brownian motion is a version of Brownian motion which makes more positive excursions from 0 than negative excursions (or vice versa). I will discuss a stochastic flow of skew Brownian motions, i.e., a family of skew Brownian motions driven by the same ``noise.'' Individual processes in the flow have a tendency to coalesce so this leads to a number of questions about the topological and stochastic nature of the flow. Lenses are pairs of distinct processes in the flow that start from the same point, diverge, and then converge.

Time:Thursday, August 26, 2004 at 3:00 pm.

Location:PDL C-36

Speaker:Wojbor A. Woyczynski (Case Western Reserve University)

Title:Critical nonlinearity exponent for Levy conservation laws

Abstract:Behavior of solutions of nonlinear evolutions of the form u_t=Lu-\nabla Nu, where L is a linear diffusive operator and N is a nonlinear operator depends on the interaqction and relative strength of L and N. In the case when L is an infinitesimal generator of a Levy process there is a critical nonlinearity which causes the asymptotic behavior of all solutions to be like the asymptotic behavior of special selfsimilar solutions. Statistical implications of this facts are indicated.

Time:Thursday, May 27, 2004 at 2:30 pm.

Location:THO 331

Speaker:Richard Sowers (University of Illinois at Urbana-Champaign)

Title:Random Perturbations of Pseudoperiodic Flows

Abstract:Arnol'd in 1991 characterized ``pseudoperiodic'' flows on the 2-dimensional torus. We consider small random perturbations of such flows. Under appropriate scaling of time, we search for an averaged picture which describes the evolution of local ``energies''. Under certain circumstances, we identify a certain limiting Markov process with glueing conditions (as suggested by Freidlin in 1996) which characterizes energy evolution.

Time:Monday, May 24, 2004 at 2:30 pm.

Location:THO 125

Speaker:Zoran Vondracek (University of Zagreb, Croatia)

Title:Ruin Probabilities for General Perturbed Risk Processes

Abstract:We study a general perturbed risk process with cumulative claims modeled by a subordinator with finite expectation, and the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs. We also give a formula that the ruin will occur by a jump of the subordinator.

Notice the different time and location

Time:Tuesday, May 18, 2004 at 2:30 pm.

Location:THO 331

Speaker:Renming Song (University of Illinois at Urbana-Champaign)

Title:Potential Theory of Geometric Stable Processes

Abstract:Geometric stable processes form a special class of Levy processes. Geometric stable processes are very useful in modeling heavy-tailed phenomena and have been used by various researchers in option pricing. In this talk we will present some recent results on the potential theory of geometric stable processes. In particular, we will talk about asymptotic behaviors of the Green function and jumping functions of geometric stable processes, and estimates on the capacities of small balls with respect to geometric stable processes. We also show that the Harnack inequality is valid for positive harmonic functions of geometric stable processes.

Time:Monday, May 10, 2004 at 2:30 pm.

Location:THO 125

Speaker:Chris Hoffman (University of Washington)

Title:Coexistence for Richardson Type Competing Spatial Growth Models

Abstract:In the Richardson growth model the vertices in Z^d can take on three possible states 0,1, and 2. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0 which are adjacent to a vertex in state 1 (or state 2) can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections while the vertices in state 0 are considered uninfected. We start the models with a single vertex in state 1 and a single vertex is in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. The key tool is applying the ergodic theorem to stationary first passage percolation.

Time:Monday, May 3, 2004 at 2:30 pm.

Location:THO 125

Speaker:Tadeusz Kulczycki (Wroclaw Technical University, visiting Purdue University)

Title:The Cauchy Process and the Steklov Problem

Abstract:Let $X_t$ be the Cauchy process in $R^d$. We investigate some of the fine properties of the semigroup of this process killed upon leaving a domain $D$. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the "Mixed Steklov Problem". Using this we derive a variational characterization for the eigenvalues of the Cauchy process in $D$. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for the Brownian motion. The results are new even in the simplest geometric setting of the interval $D=(-1,1)$ where we obtain more precise information on the size of the second eigenvalue and on the geometry of the corresponding eigenfunction.

Time:Monday, April 26, 2004 at 2:30 pm.

Location:THO 125

Speaker:Gordon Slade (University of British Columbia, Visiting Microsoft)

Title:The phase transition for random subgraphs of the n-cube

Abstract:We describe recent results, obtained in collaboration with C. Borg, J. T. Chayes, R. van der Hofstad and J. Spencer, which provide a detailed description of the phase transition for random subgraphs of the n-cube.

Time:Monday, April 19, 2004 at 2:30 pm.

Location:THO 125

Speaker:Jason Swanson (UW)

Title:The Signed Variations of the Solution to a Stochastic Heat Equation

Abstract:We will consider the solution to a certain stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time (call it $F(t)$) has the same local behavior as a fractional Brownian motion with Hurst parameter $H=1/4$. (Roughly speaking, this means that, almost surely, the function $F(t)$ is H\"{o}lder continuous with index $\alpha$ for all $\alpha<1/4$, but not for $\alpha\ge1/4$.) The function $F(t)$, therefore, has an infinite quadratic variation and, hence, is not a semimartingale. It follows then that the classical Ito calculus does not apply to $F(t)$.

Heuristic ideas about a possible new calculus for this process will be presented. These heuristics lead us in a natural way to define and study what will be called the ``signed" quadratic variation of $F(t)$. The main result to be presented is that, although the traditional quadratic variation of $F(t)$ is infinite, the signed quadratic variation converges to a Brownian motion.

Time:Monday, April 5, 2004 at 2:30 pm.

Location:THO 125

Speaker:Paul Shields

Title:Two New Methods of Markov Order Estimation

Abstract:Given an ergodic finite state Markov chain of unknown order K, the goal is to make an estimate k(n) of K from observation of a sample path of length n, such that k(n) is eventually almost sure equal to K. An important and widely used estimation method is Schwarz's BIC, which is based on Bayesian ideas. Recently, in joint work with Csiszar, BIC consistency was established without the usual prior bound assumption. The proof, however, is quite complex. I will discuss two new methods that are simple to describe, require no prior order bound, involve less computation than the BIC, and whose consistency proofs use only classical probability and information theory concepts. The first method focuses on maximum fluctuations of empirical conditional probabilities of order k over slowly growing intervals of values of k. The second method compares empirical conditional entropy of order k with an auxiliary, upwardly biased, entropy estimator. In both cases, the focused quantities have, eventually almost surely, a qualitative change in behaviour for the first time at the place where $k=K$. Extensions to random fields and to hidden Markov chains will also be discussed. (This is joint work with Yuval Peres.)

Time:Wednesday, March 24, 2004 at 2:30 pm.

Location:PDL C-36

Speaker:Marek Biskup (UCLA)

Title:Graph Distance in Long-range Percolation

Abstract:In 1967, using an ingenious sociological experiment, S. Milgram studied the length of acquaintance chains between "geometrically distant" individuals. The results led him to the famous conclusion that average two Americans are about six acquaintances (or "six handshakes") away from each other. We will model the situation in terms of long-range percolation onZ, where the nearest neighbor bonds represent the acquaintances due to geometric proximity -- people living in the house next door -- while long bonds are acquaintances established by other means -- e.g., friends from college. The question is: What is the minimal number of bonds one needs to traverse to get from site^{d}xto sitey.Thus, in addition to the usual connections between nearest neighbors on

Z, any two sites^{d}x,yinZat Euclidean distance^{d}|x-y|will be connected by an occupied bond independently with probability proportional to|x-y|, where^{-s}s>0is a parameter. UsingD(x,y)to denote the length of the shortest occupied path betweenxandy, the main question boils down to the asymptotic scaling ofD(x,y)as|x-y|tends to infinity. I will discuss a variety of possible behaviors and mention known results and open problems. Then I will sketch the proof of the fact that, whensin the interval(d,2d), the distanceD(x,y)scales like (log|x-y|)^{Delta}, where Delta^{-1}is the binary logarithm of2d/s.

SPECIAL COLLOQUIUM/PROBABILITY SEMINAR

Time:Tuesday, March 16, 2004 at 2:30 pm.

Location:SMI 205

Speaker:Wendelin Werner (Universite Paris-Sud)

Title:CONFORMAL FIELD THEORY VIA BROWNIAN LOOP SOUP

Abstract:We will (briefly) show how to relate Schramm-Loewner Evolutions and to define Conformal Field Theories using the Brownian loop-soup, a random conformally invariant countable family of overlapping Brownian loops in a domain that we introduced in a joint paper with Greg Lawler.

Time:Monday, March 8, 2004 at 2:30 pm.

Location:LOW 217

Speaker:Fadoua Balabdaoui (University of Washington)

Title:NONPARAMETRIC ESTIMATION OF A $K$-MONOTONE DENSITY ASYMPTOTIC DISTRIBUTION THEORY

Abstract:Let $k \geq 1$ be an integer. We consider nonparametric estimation of a $k$-monotone density on $(0,\infty)$ via the methods of Maximum Likelihood and Least Squares. Under the assumption that at a fixed point $x_0 > 0$, the true density $g_0$ satisfies $g^{(k)}_0(x_0) \ne 0$, we establish asymptotic minimax lower bounds for estimation of the $j$-th derivative of $g_0$ for $j=0, \cdots, k-1$. These bounds show that the rates of convergence of any estimator of $g^{(j)}_0(x_0)$ can be {\it at most} $n^{-(k-j)/(2k+1)}$. Furthermore, we are close to proving that the ML and LS estimators, $\hat{g}_n$ and $\tilde{g}_n$, achieve these rates, and that the limiting distributions depend on smooth stochastic processes $H_k$ that stay above (or below) $Y_k$, the $(k-1)$-fold integral of two-sided Brownian motion + $\left(k!/(2k)!\right) t^{2k}, t \in {\bf R}$ when $k$ is even (or odd). The key remaining difficulty for $k > 2$ consists in showing that the distance between two successive jump points $\tau^{-}_n$ and $\tau^{+}_n$ of $\hat{g}^{(k-1)}_n$ or $\tilde{g}^{(k-1)}_n$ (in the neighborhood of $x_0$) is $O_p(n^{-1/(2k+1)})$ as $n \to \infty$. Similarly, if $H_{k,c}$ denotes the envelope (or the \lq\lq invelope\rq\rq) of $Y_k$ when $k$ is odd (or even) on $\lbrack -c,c \rbrack$, $c > 0 $, it remains to prove that the distance between two successive points of touch $\tau^{-}_c$ and $\tau^{+}_c$, before and after a point $-c < t < c$, between the processes $H_{k,c}$ and $Y_k$ is $O_p(1)$ as $c \to \infty$. Several numerical examples will be shown to illustrate the theoretical results and conjectures. For that, we will introduce a $(2k-1)$-th iterative spline algorithm developed to compute the ML and LS estimators, and to obtain an approximation of the process $H_k$ on $\lbrack -c,c \rbrack $.

Joint work with Jon Wellner.

Time:Monday, March 1, 2004 at 2:30 pm.

Location:LOW 217

Speaker:Martin Barlow (University of British Columbia)

Title:RANDOM WALKS ON CRITICAL PERCOLATION CLUSTERS

Abstract:It is now known that random walks on supercritical ($p>p_c$) percolation clusters in $Z^2$ behave in many ways like the simple random walk on $Z^d$. The critical case ($p=p_c$) is much harder. One needs to consider the ``incipient infinite cluster''; in the cases where this has been defined it has a fractal structure. The easiest case is that of trees; this was studied by Kesten in 1986, but we can now revisit this problem with new techniques.

This is a joint Math Department Colloquium and Probability Seminar Talk

Time:Monday, February 23, 2004 at 2:30 pm.

Location:LOW 217

Speaker:Masatoshi Fukushima (Kansai University)

Title:POISSON POINT PROCESSES ATTACHED TO SYMMETRIC DIFFUSIONS

Abstract:Let $a$ be a non-isolated point of a topological space $S$ and $X^0$ be a symmetric diffusion on the complementary set $S_0$ which approaches to $a$ in finite time before killing with positive probability. By making use of Poisson point processes taking values in the spaces of excursions around $a$ whose characteristic measures are uniquely determined by $X^0$, we construct a symmetric diffusion on $S$ with no killing inside $S$ which extends $X^0$ on $S_0.$ We also prove that such an extension is unique in law and its resolvent and Dirichlet form admit explicit expressions in terms of $X^0.$

Time:Monday, February 9, 2004 at 2:30 pm.

Location:LOW 217

Speaker:Tilmann Gneiting (University of Washington)

Title:CONVOLUTION ROOTS OF COMPACTLY SUPPORTED RADIAL POSITIVE DEFINITE FUNCTIONS

Abstract:A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi(x) = 0$ for $|x| \geq \tau$ admits a representation of the form $$ \varphi(x) = \int u(y) \overline{u(y+x)} dy, x \in \real $$ where $u \in L_2(\real)$ is complex-valued with $u(x) = 0$ for $|x| \geq \tau/2$. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This talk examines the Boas-Kac representation under additional constraints: If $\varphi$ is real-valued and even, can the convolution root $u$ be chosen as a real-valued and/or even function?

A complete answer in terms of the zeros of the Fourier transform of $\varphi$ is obtained. Furthermore, the analogous problem for radially symmetric functions is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half support. Related results include pointwise and integral bounds on compactly supported positive definite functions and the solution to an associated minimization problem. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted.

This is joint work with Werner Ehm and Donald Richards.

Time:Monday, February 2, 2004 at 2:30 pm.

Location:LOW 217

Speaker:David Galvin (Microsoft Research)

Title:THE ``ENTROPY METHOD'' IN COMBINATORICS

Abstract:The binary entropy of a finite-range uniform random variable is exactly the log of the size of the range space. For this reason, entropy methods have become quite popular recently for tackling certain enumerative problems in combinatorics---any bounds that can be put on the entropy of a uniformly chosen member of a set correspond immediately to bounds on the size of the set.

In this talk I will introduce an entropy inequality due to J. B. Shearer that is proving to be a powerful tool in this direction. I will give two applications. The first is a swift entropy proof of an old result of Loomis and Watson, bounding the volume of a body in $R^d$ in terms the volumes of its $(d-1)$-dimensional projections. The second is a recent result (joint with P. Tetali of Georgia Tech) giving a tight upper bound on the number of graph homomorphisms from a regular bipartite graph to any fixed constraint graph.

Time:Monday, January 26, 2004 at 2:30 pm.

Location:LOW 217

Speaker:D. Brian Walton (University of Washington)

Title:HIDDEN MARKOV MODELS AND SINGLE-MOLECULE MOTOR PROTEIN EXPERIMENTS

Abstract:Motor proteins convert chemical energy into directed mechanical motion, often along filamentary tracks. This talk will introduce a particular motor protein, kinesin, which converts the energy released from ATP hydrolysis into motion along a microtubule. Recent biophysical experiments have tethered a glass bead to an individual kinesin which allows to impose an external force by laser tweezers as well as to record detailed position measurements of the bead as it moves. Using a continuous time Markov chain to model the driving cycle of the protein and a tethered diffusion to model the position of the bead, we develop a hidden Markov model for analyzing the experimental data. Such a model allows for parameter estimation and model selection.

Time:Monday, January 11, 2004 at 2:30 pm.

Location:LOW 217

Speaker:Panki Kim (University of Washington)

Title:$(-\Delta)^{\alpha/2}$-HARMONIC FUNCTIONS IN BOUNDED $\kappa$-FAT OPEN SET

Abstract:In this talk, we will discuss about the boundary behavior of $(-\Delta)^{\alpha/2}$-harmonic functions (equivalently harmonic function for symmetric $\alpha$-stable processes) in bounded $\kappa$-fat open set where $\alpha \in (0,2)$.

Consider a positive $(-\Delta)^{\alpha/2}$-harmonic function $u$ in a bounded $\kappa$-fat open set $D$, and a positive $(-\Delta)^{\alpha/2}$-harmonic function in $D$ vanishing outside of $D$. It is true that non-tangential limits of $u/h$ exist almost everywhere with respect to the Martin-representing measure of $h$.

We also study relative Fatou's theorem for operators obtained from the generator of the killed $\alpha$-stable process in bounded $\kappa$-fat open set $D$ through non-local Feynman-Kac transforms.

Time:Monday, Decemmber 8, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Alan Michael Hammond (UC Berkeley)

Title:Kinetic Limit for Coagulating and Diffusing Particles

Abstract:Consider a system of $N$ balls of radius $\epsilon$, initially placed at Poisson random locations in a compact region of space $R^d$, with $d \geq 3$. Each ball has a mass $m \in N$, and evolves as Brownian motion with a diffusion rate $d(m)$. When two balls come close enough that they overlap, they are prone to coagulate, being replaced by a new ball that combines the mass of the old two, and pursues a Brownian evolution at a new diffusion rate. In this joint work with Fraydoun Rezakhanlou, we study the macroscopic evolution of the particle densities, in the mean free path regime, where $N \epsilon^{d-2}$ tends to a constant as $N$ is taken high (this limit is chosen to ensure that the typical collision rate of a particle remains of unit order per unit time). This evolution takes the form of a coupled system of PDEs, indexed by the mass parameter, and sometimes called the discrete diffusive coagulation equations. Time permitting, we will discuss the implications of introducing fragmentation into the model.

Time:Monday, Decemmber 1, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Gabor Pete (UC Berkeley)

Title:Bootstrap percolation on infinite trees and non-amenable groups

Abstract:Consider an arbitrary infinite graph with two possible states for each vertex: {\it vacant} or {\it occupied}. We start with a random configuration in which each vertex is occupied independently with a fixed probability $p$, and then follow a deterministic {\it spreading rule}: if a vacant site has at least $k$ occupied neighbors at a certain time step, then it becomes occupied in the next step. We are interested in the values of $p$ for which complete occupation of the graph happens with positive probability. This process is well-studied on $Z^d$; here we investigate it on infinite trees and non-amenable Cayley graphs. For example, on general trees we find the following discontinuity: if the {\it branching number} of a tree is strictly less than $k$, then no $p<1$ will result in occupying the tree completely, while on the $k$-regular tree, an initial configuration with $p=1-1/k$ already succeeds almost surely.

This is joint work with J\'ozsef Balogh and Yuval Peres.

Time:Monday, November 24, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:David Revelle (UC Berkeley)

Title:Some exotic behavior of random walks on lamplighter groups

Abstract:The behavior of random walks on finitely generated groups is best understood for groups that are either non-amenable or have polynomial volume growth. Random walks on amenable groups of exponential volume growth, however, can have a wide variety of behaviors, many of which do not occur in classical examples. They gave examples of groups for which the probability of a random walk returning to the identity at time $n$ decays at a rate that cannot occur on Lie groups, and have been used to disprove conjectures about the behavior of harmonic functions on amenable groups as well as give examples of groups on which inward biased walks can escape from the origin more quickly than unbiased walks.

The first examples of groups with many of these new behaviors have been lamplighter groups and other wreath products. We will present a number of these interesting examples and discuss their place in the general theory of random walks on groups.

Time:Monday, November 17, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Benjamin J Morris (Indiana University)

Title:The mixing time for simple exclusion

Abstract:We obtain a tight bound of $O(L^2 log r)$ for the mixing time of the exclusion process in the $d$-dimensional box of side length $L$ with $r \leq \half L^d$ particles.

Time:Monday, November 10, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Jiangang Ying (Fudan University and University of Washington)

Title:Time change and Feller measure

Abstract:We will discuss an identity going back to 1931, which is now called Douglas integral. The so-called Feller measure is identified with the jumping measure of the time changed process and it is proven that the Douglas integral type formula holds for diffusion process.

Time:Monday, November 3, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Scott Sheffield (Visiting Microsoft Research)

Title:Schramm-Loewner evolution and the Gaussian free field

Abstract:Many physical phenomena---like the adherence of particles to crystal surfaces or the way crystal surfaces fluctuate---are essentially two-dimensional. The boundaries of particle clusters and the contour lines of fluctuating surfaces may be viewed asrandom non-intersecting loop ensembles.

The discreteGaussian free field(a.k.a.massless free fieldorharmonic crystal) is a popular model for describing the fluctuations of random surfaces. We show that the scaling limit of certain contour lines of discrete Gaussian free fields is given by the Schramm-Loewner evolution, SLE(4). We also introduce a conformally invariant class of random two-dimensional loop ensembles. This talk is based on joint work with Oded Schramm.

Time:Monday, October 27, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Krzysztof Burdzy (University of Washington)

Title:TRAPS FOR REFLECTED BROWNIAN MOTION

Abstract:Consider an open set $D\subset R^d$, $d\geq 2$, and a closed ball $B\subset D$. Let $\E^xT_B$ denote the expectation of the hitting time of $B$ for reflected Brownian motion in $D$ starting from $x\in D$. A set $D$ is called a trap domain if $\sup_x \E^x T_B = \infty$. One can fully characterize simply connected planar trap domains using a geometric condition. A number of less complete results for multidimensional domains are available. Time permitting, I will discuss the relationship between trap domains and some other potential theoretic properties of $D$ such as compactness of the 1-resolvent of the Neumann Laplacian. An answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain $D$ and compactness of the 1-resolvent of the Neumann Laplacian in $D$ will be given.

Joint work with Zhenqing Chen and Don Marshall.

Time:Monday, October 20, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Ping Ao (UW, Departments of Mechanical Engineering and Physics)

Title:Stochastic Differential Equations from a Scientist's View

Abstract:In physics and other branches of natural and social sciences as well as in engineering, one often encounters the solving of stochastic differential equations or Langevin equations. Among the important questions, the global trend of evolution has been dominant. In my talk I will put stochastic differential equations in the broader perspective of physics, and formulate a few mathematical problems based on a physicist's intuition: the structure of stochastic differential equations, the construction of potential hence the generalized Boltzmann-Gibbs distribution, and the connection to partial differential equations, Fokker-Planck equations. I am looking forward to professional mathematicians' feedback.

Time:Monday, October 12, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Vlada Limic (University of British Columbia)

Title:Attracting edge property for reinforced random walks

Abstract:Using martingale techniques and comparison with the generalized Urn scheme, one can show that the edge reinforced random walk on a graph of bounded degree, with the {\em weight function} W(k) = k^\rho,\,\rho > 1, crosses a random {\em attracting} edge at all large times. If the graph is a triangle, the above result is in agreement with a conjecture of Sellke (1994).

Time:Monday, October 6, 2003 at 2:30 pm.

Location:PDL C-36

Speaker:Panki Kim (University of Washington)

Title:Weak Convergence of Censored and Reflected Stable Processes

Abstract:A possible way of studying properties of some stochastic process in a open set $D$ with ``rough'' boundary is first to consider a sequence of stochastic processes in smooth open sets $D_k$ increasing to $D$. Then one could consider properties of that stochastic process in $D$ through the weak limit.

In this talk, we will discuss weak convergence of censored and reflected stable processes, which were introduced recently by Bogdan, Burdzy and Chen. One of the powerful tools in studying weak convergence of Markov processes is the so-called Mosco convergence. Some generalization of classical approximation theory (Trotter-Kato) and Mosco convergence will be introduced. We will also discuss about the Skorohod topology on the space of right continuous functions with left limits, and the tightness on that space. Some sufficient conditions for weak convergence will be discussed.

Time:Wednesday, September 24, 2003 at 2:30 pm.

Location:THO 231

Speaker:Serban Nacu (UC Berkeley)

Title:Fast Simulation of New Coins from Old

Abstract:We consider the problem of using a coin with probability of heads $p$ ($p$ unknown) to simulate a coin with probability of heads $f(p)$, where $f$ is some known function. We prove that if $f:I \rightarrow (0,1)$ is real analytic on the closed interval $I \subset (0,1)$, then it has a fast simulation (the number of needed inputs has tails which decay exponentially fast). Conversely, if a function $f$ has a fast simulation on an open set, it is real analytic on that set.

This is a joint work with with Yuval Peres.

Time:Monday, June 2, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Takashi Kumagai (Tokyo University, Japan)

Title:STABILITY OF HEAT KERNEL ESTIMATES AND HOMOGENIZATION ON FRACTAL GRAPHS

Abstract:We consider stability of the long time behaviour of the Markov process (solution of the heat equation) when local structures are modified. The talk consists of three parts:

1) We introduce several recent results (including ours) on equivalence conditions for sub-Gaussian heat kernel estimates and their stability.

2) We give heat kernel estimates for a concrete class of finitely ramified fractal graphs.

3) We consider homogenization problem on the class of fractals.

Time:Wednesday, May 28, 2003 at 2:30 pm.

Location:SAV 131

Speaker:Neal Madras (York University)

Title:DECOMPOSITION OF MARKOV CHAINS

Abstract:The speed of convergence of a Markov chain to its equilibrium distribution has been a subject of intense study in recent years. Much progress has been made for chains with nice structure, but one often has to deal with chains that are not quite so nice. This talk will describe a method that analyzes convergence rates by decomposing a Markov chain into smaller pieces. The idea is that if the chain equilibrates rapidly on each piece, and if the chain moves from piece to piece efficiently, then the entire chain equilibrates rapidly. We will give some examples of this method in action. This talk describes joint work with Dana Randall and with Zhongrong Zheng.

Time:Monday, May 19, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Jon A. Wellner (University of Washington)

Title:CONCENTRATION INEQUALITIES AND LIMIT THEOREMS FOR RATIOS

Abstract:Concentration inequalities have become increasingly valuable as tools in empirical process theory. In this talk I will discuss recent concentration inequalities due to Talagrand, Ledoux, and Massart. I will show how these can be used to derive some new inequalities for ratio-type suprema of empirical processes. The new inequalities will then be used to prove several new limit theorems for ratio-type suprema and to recover a number of the results of Alexander.

As a statistical application, I will briefly discuss an ``oracle inequality'' for nonparametric regression.

Time:Monday, May 12, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Zhen-Qing Chen (University of Washington)

Title:BROWNIAN MOTION WITH SINGULAR DRIFT

Abstract:We consider the stochastic differential equation $$dX_t=dW_t+dA_t,$$ where $W_t$ is $d$-dimensional Brownian motion with $d\geq 2$ and the $i$th component of $A_t$ is a process of bounded variation that stands in the same relationship to a measure $\pi^i$ as $\int_0^t f(X_s) ds$ does to the measure $f(x)dx$. We prove weak existence and uniqueness for the above stochastic differential equation when the measures $\pi^i$ are members of the Kato class $\K_{d-1}$. The infinitesimal generator of such a process is a Laplacian with a very singular drift $\pi \cdot \nabla$.As a typical example, we obtain a Brownian motion that has upward drift when in certain fractal-like sets, and show that such a process is unique in law.

This is a joint work with Rich Bass.

Time:Monday, May 5, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Chris Hoffman (University of Washington)

Title:PHASE TRANSITIONS IN ONE DIMENSIONAL SYSTEMS

Abstract:Many probabilistic models with short range interactions, such as percolation and the Ising model, exhibit phase transitions in two dimensions, but not in one dimension. One dimensional versions of the Ising model and percolation with long range interactions can exhibit phase transitions. I will also discuss phase transitions in processes defined by a continuous g function. I will describe an example due to Bramson and Kalikow and talk about the question of how long the interactions must be in order to have a phase transition.

This is joint work with Noam Berger and Vladas Sidoravicius.

Time:Monday, April 28, 2003 at 2:30 pm.

Location:SMI 313

Speaker:David Wilson (Microsoft Theory Group)

Title:CONFORMAL RADII OF LOOP MODELS

Abstract:We consider two loop models related to domino tilings. The first system of loops is formed by superimposing two uniformly random domino tilings, and is conjectured to be related to $SLE_4$. The second system of loops are the loops of the ``cycle-rooted spanning forest'' formed when applying the Temperleyan correspondence to a region containing a hole, and is conjectured to be related to $SLE_2$. Under the assumption of conformal invariance, and a couple of other assumptions, we derive the distribution of conformal radii of the nested loops, as well as the ``electrical thickness''', which is the difference between the conformal radius and capacity.

Joint work with Rick Kenyon.

Time:Monday, April 21, 2003 at 2:30 pm.

Location:SMI 313

Speaker:David Galvin (Microsoft Theory Group)

Title:GIBBS MEASURES FOR INDEPENDENT SET MODELS

Abstract:For a graph $G$ with vertex set $V(G)$, write $I(G)$ for the collection of {\it independent sets} of $G$ (sets of vertices no two of which are adjacent). When $G$ is finite, there is no problem interpreting the phrase ``uniformly chosen member of $I(G)$''. But how do we interpret it when $G$ (and hence $I(G)$) is infinite? One natural answer is to say that a measure $m$ on $I(G)$ is uniform if its restriction to finite patches is uniform. [Formally, this means that for $II$ chosen from $I(G)$ according to $m$, and for all finite subsets $W$ of $V(G)$, the conditional distribution of $II \cap W$ given $II \cap (V\setminus W)$ is ($m$-a.s.) uniform on the independent sets of $W$ which are compatible with $II \cap (V\setminus W)$]. We call such a measure a {\it Gibbs measure}. General arguments show that Gibbs measures always exist. For some infinite graphs, such as the two-dimensional integer lattice $Z^2$, the Gibbs measure is unique. But this is not the case for every graph. Indeed, if the dimension $d$ is large enough, there are at least two Gibbs measures on $I(Z^d)$. I will outline a proof of this result, which combines arguments from statistical physics with delicate combinatorial enumeration.

This is joint work with Jeff Kahn of Rutgers.

Time:Monday, April 14, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Federico Marchetti (Politecnico di Milano)

Title:RANDOM SEQUENCES OF NESTED INTERVALS AND THE BARGAINING PROBLEM

Abstract:The limit of random sequences of nested intervals can be thought as providing a ``dynamic'' solution to the so-called bargaining problem. We discuss a simple Markovian case. As an example it is easy to recover Nash's 1950 solution, as the mean value of such a probabilistic solution. One application is, in turn, to use the bargaining model to select a price in incomplete markets.

Mathematics Department Colloquium:

Time:Thursday, April 3, 2003 at 4 pm.

Location:Thomson 119

Speaker:Michael Roeckner (University of Bielefeld, Germany)

Title:Heat equation in infinitely many variables and applications to stochastic partial differential equations

Abstract:I will first review the connection between ordinary stochastic differential equations (OSDE) on ${\bf R}^d$ and parabolic partial differential equations (such as the classical heat equation). Subsequently, a generalization to OSDE's in infinite dimensions will be given. A purely analytic approach to solve the corresponding (generalized) heat equation in infinitely many variables will be presented. Infinite dimensional OSDE's related to parabolic stochastic partial differential equation will be discussed. Examples will include the stochastic Ginzburg-Landau equation, the generalized stochastic Burgers equation, and the stochastic Navier-Stokes equation.

Time:Tuesday, April 1, 2003 at 2:30 pm.

Location:SMI 404

Speaker:Wembo Li (University of Delaware)

Title:Recent developments on small ball probabilities for stable processes

Abstract:The small ball probability or small deviation studies the behavior of \log \mu (x : ||x|| \le t) as t tend to zero for a given measure \mu and a norm ||.||. In the literature, small ball probabilities of various types are studied and applied to many problems of interest under different names such as small deviation, lower tail behavior, two sided boundary crossing probabilities, the first exit probabilities, the asymptotes of Laplace transforms, etc. We will overview some of the recent developments for stable measures/processes and present various connections with other areas of probability and analysis.

Time:Monday, March 31, 2003 at 2:30 pm.

Location:SMI 313

Speaker:Balint Toth (Budapest University of Technology and Economics)

Title:Derivation of Leroux's pde as the hydrodynamic limit of a two component system

Abstract:By applying a stochastic version of the method of \emph{compensated compactness} (originally developed by F. Murat, L. Tartar, R. DiPerna for proving convergence of the vanishing viscosity solutions of hyperbolic systems of conservation laws) we prove hydrodynamic limit -- even beyond the appearance of shock waves -- for an interacting particle system with two conservation laws. We obtain Leroux's system as Euler equations for our models. This is the first time that hyperbolic hydrodynamic limit is proved beyond the appearence of shock wavas for systems with more than one conservation laws. Joint work with Jozsef Fritz.

Time:Monday, March 10, 2003 at 2:30 pm.

Location:MGH 242

Speaker:Christian Borgs (Microsoft Research)

Title:The Scaling Window for Percolation on Finite Transitive Graphs

Abstract:Many models of practical relevance, like faulty wireless networks, are well described by random subgraphs of finite graphs, or, more probabilistically, by percolation on finite graphs. For both theoretical and practical reasons, one of the most interesting properties of these models is the behavior of the largest connected cluster as the underlying edge density is varied.

While the behavior of the largest cluster is well understood for the complete graph, which was first systematically studied by Erdos and Renyi in 1963, not much was known for general finite graphs.

In this talk I formulate conditions under which transitive graphs on N vertices exhibit the same scaling behavior as the complete graph, i.e. a scaling window of width N^{-1/3} in which the size of the largest cluster is of order N^{2/3}.

This work is in collaboration with Jennifer Chayes, Gordon Slade, Joel Spencer and Remco van der Hofstad.

Time:Monday, March 3, 2003 at 2:30 pm.

Location:MGH 242

Speaker:Oded Schramm (Microsoft Research)

Title:Random planar triangulations

Abstract:Triangulations of the plane with n vertices fall into finitely many isomorphism classes. The number of such classes has been determined by Tutte. Let n be large and let T_n be randomly-uniformly chosen among such isomorphism classes. The geometry of T_n turns out to be very different from the geometry one gets by random constructions based on the Euclidean geometry, such as Voronoi triangulations. In a way, T_n is a "generic" metric of the sphere. Physicists have been interested in these triangulations (under the name "quantum gravity"). The purpose of the talk will be to survey some recent progress in the understanding the geometry of the uniform triangulations.

Time:Monday, February 10, 2003 at 2:30 pm.

Location:MGH 242

Speaker:Jason Swanson (University of Washington)

Title:The p-th Variation of a Brownian Martingale with an Application to Mathematical Finance

Abstract:It is well known that a continuous martingale M_t has a finite quadratic variation, which is independent of time partitions used. Moreover, the p-th variation of M_t is zero if p>2 and infinity if p<2.

For a continuous martingale M_t that is adapted to a Brownian filtration and for p other than 2, suitably rescaling the p-th variation of M_t will result in nontrivial limits. Unlike the p=2 case, however, the limit depends on the choice of the time partitions. I will discuss what the rescaling is, what the limit is, and how it depends on the time partitions.

The special case p=1 will be used to partially generalize a result of Grannan and Swindle regarding the scaled limit of transaction costs in a model of mathematical finance.

Time:Monday, February 3, 2003 at 2:30 pm.

Location:MGH 278

Speaker:Zhen-Qing Chen (University of Washington)

Title:Boundary Trace of Reflecting Brownian Motion

Abstract:Reflecting Brownian motion (RBM) can be constructed on smooth as well as on non-smooth domains. For example, RBM in a simply connected planar domain can be defined as the image under the conformal map of RBM in a unit disk after a time change. A more robust way of defining RBM in a Euclidean domain in any dimension is to use energy form or Dirichlet form method. In this talk, we will show that, under some quite general conditions, RBM has the uniform dimension property that says the Hausdorff dimension of the image of a time set under RBM doubles that of the time set. We will also present results on the Hausdorff dimension for the occupation time set of RBM on the boundary, which then gives the Hausdorff dimension for the boundary trace of RBM. The above results are applicable to many Euclidean domains with fractal-like boundaries.

This is a part of a joint work with Itai Benjamini and Steffen Rohde.

Time:Monday, January 27, 2003 at 2:30 pm.

Location:MGH 278

Speaker:David M.Mason (University of Delaware)

Title:An Exponential Inequality for the Weighted Approximation to the Uniform Empirical Process

Abstract:Mason and van Zwet (1987) obtained a refinement to the Komlos, Major, and Tusnady (1975) Brownian bridge approximation to the uniform empirical process. From this they derived a weighted approximation to this process, which has shown itself to have some important applications in large sample theory. We will show that their refinement, in fact, leads to a much stronger result, which should be even more useful than their original weighted approximation. We demonstrate its potential applications through several interesting examples. These include a moment bound result which is useful in the study of central limit theorems for the both the trimmed and untrimmed Wasserstein distance between the empirical and the true distribution.

Time:Monday, January 13, 2003 at 2:30 pm.

Location:MGH 278

Speaker:Krzysztof Burdzy (University of Washington)

Title:LENSES IN SKEW BROWNIAN FLOW

Abstract:I will discuss a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. Due to the lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, i.e., pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. I will present qualitative and quantitative (distributional) results on the geometry of the flow and lenses.

This is joint work with Haya Kaspi, and is based on two earlier papers, one joint with Barlow, Kaspi and Mandelbaum, and the other joint with Chen.

Time:Monday, December 9, 2002 at 2:30 pm.

Location:THO 331

Speaker:Hong Qian (University of Washington)

Title:NON-EQUILIBRIUM STATISTICAL MECHANICS OF SINGLE MOLECULES

Abstract:I shall discuss the general stochastic theory of Kurchan-Lebowitz-Spohn [1] and Jarzynski [2] for non-equilibrium steady-state, and then show how to experimentally verify some of the results from single molecule experiments on enzymes. A renewal process model will be proposed for enzyme kinetics.

[1] J.L. Lebowitz and H. Spohn, J. Stat. Phys.95(1999), 333--365[2] Jarzynski, Phys. Rev. Lett.78, 2690-2693 (1997)

Time:Monday, December 2, 2002 at 2:30 pm.

Location:THO 331

Speaker:Bruce Erickson (University of Washington)

Title:THE STRONG LAW FOR RANDOMIZED RANDOM WALKS

Abstract:Abstract: I will discuss some joint work, in progress, with R. Maller on the almost sure behavior of $\{S_{T_n}/n\}$, $n\to\infty$, where ${T_n\}$ is an independent integer renewal process and $\{S_m;\, m\ge 0\}$ is a random walk on the line. Interesting (and un-intuitive, I hope) examples occur when first moments do not exists.

Time:Monday, November 25, 2002 at 2:30 pm.

Location:THO 331

Speaker:Assaf Naor (Microsoft Research)

Title:ENTROPY PRODUCTION AND THE BRUNN-MINKOWSKI INEQUALITY

Abstract:Abstract: Let $X$ be a real valued random variable with density $f$. Its entropy is defined by $\Ent(X)=-\int f\log f$. A classical inequality of Shannon and Stam states that if $X, Y$ are IID then $\Ent((X+Y)/\sqrt{2})\ge \Ent(X)$. The problem whether or not the sequence $E_n=\Ent((X_1+...+X_n)/\sqrt{n})$ is increasing for $X_1,..., X_n$ IID remained open (in particular it wasn't known whether it is always the case that $E_3\ge E_2$). In this talk we will discuss the recent positive solution of this problem due to S. Artstein, K. Ball, F. Barthe and the speaker. The proof is based on a new formula for the entropy of a marginal which is motivated by (the proof of) the Brunn-Minkowski inequality. We will also discuss several other applications of this formula. In particular, we will show that if $X$ satisfies the Poincare inequality and $Y$ is an IID copy of $X$ then: $$\Ent((X+Y)/\sqrt{2})-\Ent(X)\ge c(\Ent(G)-\Ent(X))$$ where $G$ is a Gaussian random variable with the same variance as $X$ and $c$ depends only on the Poincare constant of $X$. Since the Gaussian has maximal entropy, this strengthening of the Shannon-Stam inequality yields a quantitative information theoretic proof of the central limit theorem. Time permitting, we will also discuss higher dimensional versions of these results, as well as related inequalities such as Shannon's entropy power inequality.

Time:Monday, November 18, 2002 at 2:30 pm.

Location:THO 331

Speaker:Benjamin Morris (University of California, Berkeley)

Title:EVOLVING SETS AND MIXING

Abstract:We introduce a probabilistic technique that yields the sharpest bounds obtained on mixing times for Markov chains in terms of isoperimetric properties of the state space. We show that the bounds for mixing time in total variation obtained by Lovasz and Kannan can be refined to apply to the maximum relative deviation $|p^n(x,y)/\pi(y) -1|$ of the distribution at time $n$ from the stationary distribution $\pi$. Joint work with Yuval Peres.

Time:Monday, November 4, 2002 at 2:30 pm.

Location:THO 331

Speaker:Panki Kim (University of Washington)

Title:STABILITY OF MARTIN BOUNDARY UNDER NON-LOCAL FEYNMAN-KAC PERTURBATIONS

Abstract:For $\alpha \in (1, 2)$, a censored $\alpha$-stable process $Y$ in bounded $C^{1,1}$ open set $D$ is a process obtained from a symmetric $\alpha$-stable Levy process by restricting its Levy measure to $D$. Recently It is shown that the Martin boundary and minimal Martin boundary of $Y$ can all be identified with the Euclidean boundary of $D$. In this talk, We discuss the stability of Martin boundary and Martin representation under non-local Feynman-Kac perturbations. As an application, Fatou's Theorem for operators obtained from the generator of $Y$ through non-local Feynman-Kac perturbations will be discussed.

This talk is based on a joint paper with Z.-Q. Chen ('02).

Time:Monday, October 14, 2002 at 2:30 pm.

Location:THO 331

Speaker:Zenghu Li (Beijing Normal University and University of Oregon)

Title:STOCHASTIC INTEGRAL EQUATIONS AND MEASURE-VALUED DIFFUSIONS CARRIED BY STOCHASTIC FLOWS

Abstract:Let $\{W(ds,dy): s\ge 0, y\in\IR\}$ be a time-space white noise and $h$ a smooth, square-integrable function on $\IR$. A stochastic flow $\{x(a,t): a\in\IR, t\ge 0\}$ is defined by the equation $$ x(a,t) = a + \int_0^t\int_{\IR} h(y-x(a,s))W(ds,dy). $$ A stochastic integral equation carried by the flow is discussed, which involves a Poisson process on the space of one-dimensional excursions. We show that the equation has a pathwise unique solution, which is then a measure-valued {\it diffusion} process. The approach is of interest since the uniqueness of solution of the corresponding martingale problem still remains open. The diffusion process is interpreted as a measure-valued branching process with immigration carried by the flow $\{x(a,t): a\in\IR, t\ge 0\}$. When $\IR$ shrinks to a single point, our equation gives a decomposition of a class of one-dimensional diffusions into excursions, which is also a new result.

This talk is based on joint papers with D.A. Dawson ('02) and Z.F. Fu ('02).

Time:Monday, October 14, 2002 at 2:30 pm.

Location:THO 331

Speaker:Zhen-Qing Chen (University of Washington)

Title:HEAT KERNEL ESTIMATES FOR STABLE-LIKE PROCESSES ON $d$-SETS

Abstract:The notion of $d$-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of $d$-sets. In this talk, we will discuss stable-like processes on $d$-sets, which include reflected stable processes in Euclidean domains as a special case. More precisely, we will present parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the range of stable-like processes will also be given.

This talk is based on a recent joint work with Takashi Kumagai.

Time:Monday, October 7, 2002 at 2:30 pm.

Location:THO 331

Speaker:David Brian Walton (University of Washington)

Title:ASYMPTOTIC LAW OF INDIRECT OBSERVATIONS WITH STATE-DEPENDENT NOISE

Abstract:A little over a year ago, a paper entitled "Noise Suppression by Noise" appeared in Phys. Rev. Lett. (Vilar and Rubi, 2001). Essentially, a hidden state evolves as an Ornstein-Uhlenbeck process X(t), and observations are made of a function of the state in addition to state-dependent white noise: dY(t)=f(X(t)) dt + \beta(X(t)) dW(t). The results of Vilar and Rubi consider a perturbative and asymptotic result to the autocorrelation function, and show that the noise scale of the resulting autocorrelation function can be reduced by increasing the noise scale of the hidden state process. I will demonstrate that the result actually relates to a central limit theorem of the observation process, and I will give exact formulas for the original perturbative results in terms of appropriate expectations.

Time:Tuesday July 23, 2002 at 2:30 pm.

Location:THO 325

Speaker:Tzong-Yow Lee (University of Maryland at College Park)

Title:Integrated Brownian motions and Sturmian theory

Abstract:We shall relate various integrated Brownian motions to higher order elliptic operators, and use this connection to obtain a stochastic domination result for the integrated BM's. This is a work in progress with F. Gao, L. Greenberg, J. Hannig, and F. Torcaso.

Time:Wednesday, July 3, 2002 at 2:30 pm.

Location:Communications 230

Speaker:Siva Athreya (Indian Statistical Institute)

Title:LONG TERM BEHAVIOUR OF BROWNIAN FLOW WITH JUMPS

Abstract:We consider a stochastic jump flow in an interval $(-a,b)$, where $a,b > 0$. Each particle of the flow performs a canonical Brownian motion and jumps to zero when it reaches $-a$ or $b$. We study the long term behavior of a random measure $\mu_t$ which is the image of the flow. This is joint work with Elena Kosegyna and Steve Tanner.

Time:Monday, June 3, 2002 at 2:30 pm.

Location:Raitt Hall 107

Speaker:Noam Berger (U. of California at Berkeley and Microsoft Research)

Title:Biased random walk on percolation clusters

Abstract:We study the behavior of the biased random walk on the infinite cluster of percolation in Z^2 (with high enough parameter). We get the seemingly surprising result that when the bias is big, the speed is zero while when the bias is small, the speed is positive. This is joint work with Nina Gantert and Yuval Peres.

Time:Monday, May 20, 2002 at 2:30 pm.

Location:Raitt Hall 107

Speaker:Renming Song (University of Illinois at Urbana-Champaign)

Title:Potential theory of subordinate killed Brownian motion in a domain

Abstract:Subordination of a killed Brownian motion in a domain $D\subset \R^d$ via an $\alpha/2$-stable subordinator gives a process $Z_t$ whose infinitesimal generator is $-(-\Delta|_D)^{\alpha/2}$, the fractional power of the negative Dirichlet Laplacian. In this talk we will present some results with Z. Vondracek on this process. We study the properties of the process $Z_t$ in a Lipschitz domain $D$ by comparing the process with the rotationally invariant $\alpha$-stable process killed upon exiting $D$. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of $Z_t$, and, in the case when $D$ is a bounded $C^{1,1}$ domain, obtain bounds on the Green function and the jumping kernel of $Z_t$.

Time:Monday, May 13, 2002 at 2:30 pm.

Location:Raitt Hall 107

Speaker:Zhen-Qing Chen (University of Washington)

Title:Conditional gauge theorems and their characterizations

Abstract:Given a generator L of a Markov process X and a potential q, its associated conditional gauge function is the ratio of the Green functions for L+q and L respectively. The conditional gauge function can be represented probabilistically in terms of the conditional process of X and the Feynman-Kac functional of q. Conditional gauge theorem (CGT) says that under suitable conditions on L and q, the conditional gauge function is either bounded between two positive constants or identically infinity. CGT has been established for Laplacian (or equivalently for Brownian motion) in the 80's and for fractional Laplacian (or symmetric stable processes) in 97'.

In this talk I will present some recent progress in establishing CGT for (non-local) Feynman-Kac transforms for a large class of Markov processes. Results on analytic characterizations for the conditional gauge function to be bounded will be reported.

Time:Wednesday, May 8, 2001, 2002 at 2:30 p.m.

Location:Smith 305

Speaker:Elchanan Mossel (Microsoft Research)

Title:Some problems from biology

Abstract:I will discuss some probabilistic problems from biology that are mostly unstudied. I plan to discuss the following problems:

1. Determining keys sites in genes--Is Fourier analysis useful for non-product spaces?

2. A problem from phylogeny--how well do we understand branching processes?

3. Iterated prisoner dilemmas--can we prove that it's better to cooporate also on the lattice?

4. Crossover in genes and mixing times--a new card shuffling problem from biology.

Time:Monday, April 29, 2002 at 2:30 p.m.

Location:Raitt Hall 107

Speaker:Bela Bollobas (U. of Memphis and U. of Cambridge, visiting Microsoft Research)

Title:Bootstrap percolation on finite graphs

Abstract:A bootstrap percolation on a finite graph $G=(V,E)$ with (neighborhood) parameter $\ell$ is a nested sequence of subsets of $V$, $V_0 \subset V_1 \subset \dots $, such that for $t>0$ a vertex $v\in V$ belongs to $V_t$ iff either $v\in V_{t-1}$ or $v$ has at least $\ell$ neighbours in $V_{t-1}$. The set $V_t$ is the set of {\it sites occupied at time} $t$. Note that the entire percolation is determined by $V_0$, the set of sites occupied at time $0$. If eventually every site is occupied then we say that the starting set $V_0$ percolates. Choosing the vertices of $V_0$ at random, with probability $p$, we get a random bootstrap percolation with parameter $\ell$ and probability $p$. One of the main objects of study is the critical probability $p_c$, below which percolation is unlikely and above which it is very likely. In the talk, I shall review a number of the known results and shall report on my work in progress with Jozsef Balogh.

Time:Monday, April 22, 2002 at 2:30 p.m.

Location:Raitt Hall 107

Speaker:Krzysztof Burdzy (University of Washington)

Title:NEUMANN EIGENFUNCTIONS IN LIP DOMAINS

Abstract:A "lip domain" is a planar domain between the graphs of two Lipschitz functions whose Lipschitz constant is strictly less than 1. The second eigenvalue for the Laplacian in a lip domain with Neumann boundary conditions (i.e., the first non-zero eigenvalue) is simple. This implies that the strongest version of the "hot spots" conjecture holds for lip domains. Two conjectures of Jerison and Nadirashvili are special cases of the main result. This is joint work with Rami Atar.

Mathematics Department Colloquium:

Time:Tuesday, April 16, 2002 at 4:00 p.m.

Location:PDL C-36

Speaker:Weian Zheng (University of California, Irvine)

Title:Rate of Convergence in Homogenization of Parabolic PDEs

Abstract:We consider the solutions to ${\partial \over \partialt}u^{(n)}=a^{(n)}(x)\Delta u^{(n)}$ where $\{a^{(n)}(x)\}_{n=1,2,...}$ are random fields satisfying a ``well-mixing" condition (which is different to the usual ``strong mixing" condition). We estimate in this paper the rate of convergence of $u^{(n)}$ to the solution of a Laplace equation. Since our equation is of simple form, we get quite strong result which covers the previous homogenization results obtained on this equation.

Time:Monday, April 15, 2002 at 2:30 p.m.

Location:Raitt Hall 107

Speaker:Weian Zheng (University of California, Irvine)

Title:Discretizing a Backward Stochastic Differential Equation

Abstract:Given a probability space $(\Omega , F,P).$ Let $W_t$ be a standard Brownian motion with $(F_t)\subset F$ as its natural filtration. Given any positive constant $T<\infty $ and a random variable $\xi \in F_T$. A backward stochastic differential equation is the equation $$ Y_t = \xi - \int_t^T f(Y_s,Z_s,s) ds -\int_t^T Z_s dW_s , $$ $$ where $Y_t$ and $Z_t$ are unknown predictable processes. We give a discretization approach to solve the above equation numerically. It is known that the solution to a backward stochastic PDE is related to a semi-linear deterministic PDE. Therefore our method also give a probability method to solve a class of semi-linear PDE.

Time:Monday, April 8, 2002 at 2:30 p.m.

Location:Raitt Hall 107

Speaker:Fabio Machado (Microsoft Research and University of Sao Paulo, Brazil)

Title:Phase transition for the frog model

Abstract:We study a system of simple random walks on graphs, known as {\it frog model}. This model can be described as follows: There are active and sleeping particles living on a graph~$G$. Each active particle performs a simple random walk at discrete time. Each active particle dies after a random life time $ T $. When an active particle hits a sleeping particle, the latter becomes active. Active particles move independently. At time zero there is only one active particle on $G$ placed at its root. Phase transition results and asymptotic values for critical parameters are presented for $Z^d$ and regular trees.

Time:Wednesday, March 13, 2002 at 2:30 p.m.

Location:THO 231

Speaker:Laszlo Lovasz (Microsoft Research)

Title:MINIMA OF RANDOM LINEAR FUNCTIONS

Abstract:Let us assign independent, exponentially distributed random edge lengths to the edges of an undirected graph. Lyons, Pemantle and Peres proved that the expected length of the shortest path between two given nodes is bounded from below by the resistance between these nodes, where the resistance of an edge is the expectation of its length. This inequality can be formulated as follows: the expected length of a shortest path between two given nodes is the expected minimum of a stochastic linear program over a flow polytope, while the resistance is the minimum of a convex quadratic function over the same polytope. The inequality between these quantities holds true for an arbitrary polytope provided its blocker has integral vertices. There are several other combinatorial polytopes for which the convex quadratic function has a combinatorial meaning: for example, it can be related to hitting times of random walks and to modeling trafic congestion.

Time:Wednesday, March 6, 2002 at 2:30 p.m.

Location:MLR 316

Speaker:Alan Stacey (Cambridge University)

Title:PARTIAL IMMUNIZATION PROCESSES

Abstract:Partial Immunization Processes generalize the well-known contact process. The rate at which a site is infected can depend on whether or not the site has been previously infected, representing the fact that once a site has been infected and recovered it may be somewhat immunized against future infection or, perhaps, may be more susceptible to future infection. These processes can exhibit a phase of weak survival -- where the process survives but drifts off to infinity -- even on lattices where the contact process does not exhibit such behaviour. A conjecture is proved characterizing the strong survival phase on hypercubic lattices. Processes on trees are also studied and the phase diagram turns out to be rather rich; proofs of results on trees utilize a detailed analysis of the behaviour of the contact process on finite trees.

Time:Monday, February 25 at 2:30 p.m.

Location:MEB 237

Speaker:Itai Benjamini (Weizmann Institute and Microsoft Research)

Title:GLOBAL INFORMATION FROM LOCAL OBSERVATIONS

Abstract:We will discuss a few examples in which random processes are used to extract information on the underlying structure.

Time:Wednesday, February 20, 2002 at 2:30 p.m.

Location:THO 231

Speaker:Paul Shields

Title:RECURRENCE REVISITED

Abstract:The time for an initial k-block to appear again in an ergodic, finite-alphabet process and related waiting time problems will be discussed. I will review results I talked about a couple of years ago for the general case, then discuss the problem of when such recurrence (and or waiting) times have an asypmtotic exponential distribution, with a focus on recent results of Miguel Abadi and some conjectures.

Time:Monday, February 11, 2002 at 2:30 p.m.

Location:MEB 237

Speaker:Chris Hoffman (University of Washington)

Title:MIXING TIME FOR BIASED CARD SHUFFLING

Abstract:Consider a deck of $n$ cards labeled 1 through $n$. We employ the following biased shuffling. At each stage we pick a pair of adjacent cards uniformly at random. Then with probability $p>.5$ we replace the cards with the lower numbered card before the higher numbered card. With probability $1-p$ we replace the cards with the higher numbered card first. We prove that the mixing time for this system is $O(n^2)$. This proves a conjecture of Diaconis and Ram. This is joint work with Itai Benjamini, Noam Berger, and Elchanan Mossel.

Time:Monday, February 4, 2002 at 2:30 p.m.

Location:MEB 237

Speaker:Yevgeniy Kovchegov (Stanford University)

Title:BROWNIAN BRIDGE IN PERCOLATION AND RELATED PROCESSES

Abstract:Consider a $d$-dimensional model of a sub-critical bond percolation and a point $(a_1,a_2,...,a_d)$ in $Z^d$. The behavior of the cluster $C_a$ connecting points $(0,...,0)$ and $n(a_1,...,a_d)$ conditioned on the two points being connected is studied. An interesting result linking the asymptotic of the cluster $C_a$ (as $n\to\infty$) to that of Brownian Bridge is discovered.

Time:Monday, January 28, 2002 at 2:30 p.m.

Location:MEB 237

Speaker:Bruce Erickson (University of Washington)

Title:EXISTENCE OF IMPPROPER INTEGRALS WITH LEVY INTEGRATORS AND EXPONENTIAL LEVY INTEGRANDS

Abstract:We determine conditions under which an improper stochastic integral $\int_0^\infty f_s \exp(-\xi_{s-})d\eta_s $ converges, a.s., where $(\xi,\eta)$ is a two dimensional L\'{e}vy process and $f$ is a bounded non-anticipating functional of (\xi,\eta). In the case $f = 1$ we show that our conditions become necessary if in addition the support of the two-dimensional L\'{e}vy measure of the process does not lie in a one-dimensional curve of the form $\{(x,y) : y + k\exp(-x) = k \}$ for any real number $k$.

Time:Wednesday, January 23, 2002 at 2:30 p.m.

Location:THO 231

Speaker:Fabio Martinelli (University of Rome)

Title:RELAXATION TIME FOR ASYMMETRIC SIMPLE EXLUSION MODELS

Abstract:We analyze the relaxation time (inverse of the spectral gap) for a class of reversible asymmetric simple exclusion models. The latter can be viewed as n independent simple random walks with drift in some finite subgraph of $Z^d$ that avoid each other. We will also briefly mention the connection of the above problem with certain quantum spin models. Joint work with P. Caputo (Rome).

Time:Monday, January 14, 2002 at 2:30 p.m.

Location:MEB 237

Speaker:Takashi Kumagai (Kyoto University)

Title:FUNCTION SPACES AND STOCHASTIC PROCESSES ON FRACTALS

Abstract:Since the late 80's of the last century, there has been a lot of development in the mathematical study of stochastic processes and the corresponding operators on fractals. On the other hand, there has been intensive study of Besov spaces, (which are roughly speaking, fractional extensions of Sobolev spaces) on $d$-sets, which correspond to regular fractals. In this talk, we summarize recent work to connect these two areas and introduce local and non-local Dirichlet forms whose domains are Besov spaces. As an application, we will introduce diffusion processes on fractal fields, a collection of fractals, of in general different Hausdorff dimensions, embedded in Euclidean spaces.

Time:Wednesday, January 9, 2002 at 2:30 p.m.

Location:Smith 304

Speaker:Bernard Sapoval (Ecole Polytechnique, Palaiseau)

Title:LAPLACIAN TRANSPORT TO AND ACROSS IRREGULAR SURFACES: FROM ELECTRODES TO MAMMALIAN LUNGS

Abstract:Several phenomena in physics, chemistry and biology, can be brought to a single mathematical problem: transport across a "resistive" irregular surface driven by Laplacian fields. This includes charge transfer in batteries, species transfer in catalysts or molecular transfer across membranes. The mathematical problem is that of the properties of the solutions of Laplace equation with Fourier (or Robin) boundary conditions. We give an approximate way to treat this question and compare with experiments. We then introduce the more fundamental mathematical object, namely the Brownian self-transport operator, which governs this type of phenomenon.

Time:Monday, December 3, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Panki Kim (UW)

Title:Green Function Estimates and Martin Boundary of Censored Stable Processes

Abstract:For $\alpha \in (0, 2)$, a censored $\alpha$-stable process $Y$ in an open set $D$ is a process obtained from a symmetric $\alpha$-stable L\'evy process by restricting its L\'evy measure to $D$. It is recently known that when $D$ is a bounded Lipschitz open set, $Y$ is transient if and only if $\alpha>1$. In this talk, we will present sharp two-sided estimates for Green functions of censored $\alpha$ stable process $Y$ in a bounded $C^{1,1}$ open set $D$ for $1<\alpha<2$. The Martin boundary theory for $Y$ and for processes obtained from $Y$ through Feynman-Kac transformations with discontinuous additive functionals will also be discussed.

This talk is based on a joint work with Zhen-Qing Chen.

Time:Monday, November 26, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:David Wilson (Miscrosoft Research)

Title:Critical Resonance in the Non-intersecting Lattice Path Model

Abstract:We study the phase transition in the honeycomb dimer model (equivalently, monotone non-intersecting lattice path model). At the critical point the system has a strong long-range dependence; in particular, periodic boundary conditions give rise to a ``resonance'' phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain. Joint work with Richard Kenyon.

Time:Monday, November 19, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Youngmee Kwon (UW and Hansung University, Korea)

Title:On the blow up phenomena before crash

Abstract:We consider jump processes as bubble and crash models of stock price. The jumps of the processes are interpreted as crashes and we assume that there are only downward jumps with size a fixed fraction of the current price. Moreover, we assume that the jump intensity is a nondecreasing function of the current price or the bubble size, say $\lambda(p)$ ($p=p(t)$: price). Through this setting, we are able to put the endogenous variables into consideration.

For the case of $\lambda(p)=p^{\alpha}, \alpha>0$, we show that the price $p$ should explode in finite time, say $t_e$, conditional on no crash. This implies that the crash should occur before $t_e$, since the jump intensity tends to infinity as $p$ explodes. Also, for the case of $\lambda(p)=(\ln p)^{\alpha}$, we show that $\al =1$ is the borderline of two different classes of the processes. In fact, the price $p$ explodes in finite time if $\alpha>1$ and grows super exponentially but never explodes in finite time if $0<\alpha\leq 1$. We generalize the model by adding Brownian noise and examine the blow up properties of the sample paths. It turns out that the noise has much influence on determining the crash time. Finally, we calculate the crash time distribution of the above endogenous models explicitly.

Time:Monday, November 5, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Steffen Rohde (UW)

Title:Basic properties of SLE

Abstract:This talk is about Oded Schramm's Stochastic Loewner Evolution SLE, which can be thought of as a random family of compact sets in the plane. SLE has played a crucial role in some recent spectacular results in probability theory, most notably the proof of Mandelbrots conjecture about the dimension of the Brownian frontier (Lawler-Schramm-Werner) and Smirnov's result about scaling limits of critical percolation. In this talk, I will review definitions, some of these results, and then discuss some path properties of SLE (joint work with Oded Schramm).

Mathematics Department Colloquium

Time:Tuesday, October 23, 2000 at 4:00 p.m.

Location:PDL C-36

Speaker:Rodrigo Banuelos (Purdue University)

Title:Generalized Isoperimetric Inequalities

Abstract:The rearrangement inequalities for multiple integrals of H.S. Brascamp, E.H. Lieb, and J.M. Luttinger provide a powerful and elegant method for proving many of the classical geometric and physical isoperimetric inequalities for regions in Euclidean space. These include, amongst others, the classical isoperimetric inequality, the Rayleigh--Faber--Krahn inequality for the lowest eigenvalue of regions of fixed volume, isoperimetric inequalities for the trace of the Dirichlet heat kernel, and the Polya--Szego isoperimetric inequality for electrostatic capacity. After discussing some of these classical results, we will present new versions of multiple integral inequalities from which other ``generalized" isoperimetric inequalities for heat kernels of Schrodinger operators follow. Besides being of independent interest, such ioperimetric inequalities for heat kernels imply sharp inequalities for the lowest eigenvalue and the spectral gap of the Dirichlet Laplacian in certain convex regions of fixed diameter and fixed inradius (radius of largest ball in the region). In particular, these results improve the spectral gap bounds of I.M. Singer--B.Wang--S.T.Yau-- S.S.T.Yau and prove some special cases of a conjecture of M. van den Berg (Problem #44 in Yau's 1990 ``open problems in geometry") on the size of the spectral gap.

This talk is particularly designed for a general audience. We will discuss results, show some pictures, but provide as few technicalities as possible.

Time:Monday, October 22, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Rodrigo Banuelos (Purdue University)

Title:Space-Time Brownian martingales and $L^p$ Inequalities for $\partial$ and ${\overline \partial}$

Abstract:We will discuss some applications of martingales to some open problems concerning $L^p$ inequalities between $\partial f$ and ${\overline\partial} f$ and their connections to singular integral operators. More precisely, consider the following conjecture of T. Iwaniec (1982): Let $1 < p < \infty$ and set $p^*=\max\{p, q\}$ where $q$ is the conjugate exponent of $p$. For $f\in W^{1, p}(C, C)$ the following inequality should be true:

$$ \|{\overline\partial} f\|_p\leq (p^*-1)\|{\partial} f\|_p. (*} $$

This conjecture has been of considerable interest because of its many applications to quasiconformal mappings and to the regularity of solutions to some nonlinear PDE's. The conjecture is equivalent to an estimate on the norm of the Beurling-Ahlfors operator (a singular integral in the complex plane). Until recently the best known estimates for (*) were those obtained by the speaker and G. Wang ($\|{\overline\partial} f\|_p\leq 4(p^*-1)\|{\partial} f\|_p $) in 1995 using martingale inequalities and stochastic integration. Recently (in 2000) Nazarov and Volberg improved the estimate by a factor of 2. In this talk we will explain how the same stochastic analysis techniques used in the paper with Wang can be use to obtain the Nazarov-Volberg estimate and to provide some additional information on this conjecture.

Time:Monday, October 15, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Ross Maller (University of Western Australia)

Title:Stability of Perpetuities

Abstract:For a series of randomly discounted terms (a generalised ``perpetuity'') we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to $\infty$. These turn out to be the only possible forms of asymptotic behaviour. This settles the existence problem for a one-dimensional perpetuity, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one. As an application, conditions for stability of the ARCH(1) and GARCH(1,1) processes used in financial data modelling will be given. Some possible extensions (Levy Processes) will be briefly outlined.

Time:Monday, October 8, 2001 at 2:30 p.m.

Location:SMI 307

Speaker:Zhen-Qing Chen (University of Washington)

Title:Censored Stable Processes

Abstract:In this talk we will discuss a new topic in boundary potential theory for discontinuous Levy processes. We will present several constructions of a ``censored stable process'' in an open set $D$ in $R^n$, i.e., a symmetric stable process which is not allowed to jump outside $D$. We will address the question of whether the process will approach the boundary of $D$ in a finite time. Sharp conditions will be given for such boundary approach in terms of the stability index $\alpha$ and the ``thickness'' of the boundary. As a corollary, new results are obtained concerning Besov spaces on non-smooth domains, including the critical exponent case. If time permits, a boundary Harnack principle in $C^{1,1}$ open sets will also be presented, as well as results about the decay rate of the corresponding harmonic functions which vanish on a part of the boundary.

This talk is based on a joint work with K. Bogdan and K. Burdzy.

Time:Monday, September 10, 2001 at 2:30 p.m.

Location:Padelford C401

Speaker:Yukio Ogura (Saga University, Japan)

Title:Completion of a Class of One-dimensional Diffusion Processes

Abstract:We are concerned with a class of one-dimensional diffusion processes (ODP) on an interval $I$ associated with the Dirichlet forms ${\cal E}(u,u) =\int_I|u'|^2 \varphi dx$ on $L^2(\varphi dx)$ with $\varphi\in C^\infty(I)$ (in other words, distorted processes), and seek a condition for its precompactness in some sense. An elemet of the closure is no longer an ODP in general, but is a system of bi-generalized diffusion processes which were introduced by the speaker in 1989. There are abundant examples including the class of $\{\varphi_n\}$ of positive $C^\infty$ functions on $I=[0,l]$ such that $\varphi_n(x) = 1$ for $x\in [0,a_{0,n}]\cup [b_{k,n},l]$, $x = a_{i,n}$ or $x = b_{i,n}$, $\lim_n\int_{a_{i,n}}^{b_{i,n}}\varphi_n dx =\alpha_i\in(0,\infty]$, and $\lim_n\int_{b_{i,n}}^{a_{i+1,n}} 1/\varphi_n dx =\beta_i\in(0,\infty]$, for $i=0,1,\ldots,n-1$, where $0This is a joint work with Takashi Shioya and Matsuyo Tomisaki.

Time:Monday, August 27, 2001 at 2:30 p.m.

Location:Padelford C401

Speaker:Zhi-Ming Ma (Academia Sinica, Beijing)

Title:SOME RECENT RESULTS ON PROBEBILITY THEORY

Abstract:In this talk I present some new results on Probability Theory in which I have recently been involved. It will involve the following topics.Dirichlet forms and stochastic analysis on configuation spaces.

The property of Markove processes in Shiga model of infinite particle systems.

The study of uncountable independent random variables and its application to economic systems.

A new type of measure valued processes associated with stochastic flows

Time:Wednesday, May 30, 2001 at 2:30 p.m.

Location:RAI 109

Speaker:Itai Benjamini (Weizmann Institute and Microsoft Research)

Title:PERCOLATION ON GRAPHS

Abstract:A review of percolation on graphs will be presented. One point is to see how coarse geometric properties of the underling graphs manifests themself in the behaviour of the percolation process. Several problems will be mentioned.

Time:Monday, May 21, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Ron Getoor (University of California, San Diego)

Title:GENERATORS AND SCHRODINGER EQUATIONS ASSOCIATED WITH MARKOV PROCESSES

Abstract:Let G be the generator of a nice Markov process. Then

(*) (G+p)u = f

is called the Schroedinger equation with potential p. If p and f are reasonable functions a solution u may be expressed using the Feynman-Kac formula. Recently there has been much interest in the situation in which p is a measure. It then seems natural to extend the domain of G so that it maps functions to measures and interpret (*) as an equation between measures where the right hand side is regarded as fm with m a given background measure--Lebesgue measure in classical cases. In this talk I will try to motivate and describe informally with as few technicalities as possible the definition of the extended generator. I will indicate some results for (*) and discuss the case f = 0 when the solutions are called p-harmonic.

Time:Monday, May 14, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Siva Athreya (University of British Columbia)

Title:UNIQUENESS FOR DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS AND SUPER-MARKOV CHAINS

Abstract:We will consider diffusions corresponding to the generator $$ L f(x) = \sum_{i=1}^d x_i \gamma_i(x)\frac{\partial^2} {\partial{x_i}^2}f(x) + b_i(x) \frac{\partial}{\partial{x_i}}f(x),$$ $x \in [0,\infty)^d,$ for continuous $\gamma_i, b_i : [0,\infty)^d \rightarrow \bR$ with $\gamma_i$ nonnegative. Our aim will be to establish uniqueness for the corresponding martingale problem under certain non-degeneracy conditions on $b_i, \gamma_i$. We will begin by briefly explaining the motivation for studying such diffusions, followed by a discussion on why standard techniques fail and conclude with a brief description of the proof.

This is joint work with Martin Barlow, Richard Bass, and Edwin Perkins.

Time:Wednesday, May 7, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Zhenqing Chen (University of Washington)

Title:DRIFT TRANSFORMATION AND GREEN FUNCTION ESTIMATE FOR DISCONTINUOUS PROCESSES

Abstract:In this talk we will discuss processes obtained from symmetric stable processes in bounded domains through pure jump Girsanov transform and Feynman-Kac transform, which include relativistic stable processes. We show that when pure jump transform does not amplify small jumps of stable process too much and the potential in Feynman-Kac transform is not too large, the Green function of perturbed process is comparable to that of symmetric stable process. This result is obtained by first establishing a conditional gauge theorem for discontinuous additive functionals, and is in fact valid for more general processes. As an application, we obtain that the Green functions for the relativistic stable processes in bounded smooth domains are comparable to that of symmetric stable processes, which recovers the recent result of Ryznar.

Time:Wednesday, May 2, 2001 at 2:30 p.m.

Location:RAI 109

Speaker:Krzysztof Burdzy (University of Washington)

Title:BROWNIAN MOTION REFLECTED ON BROWNIAN MOTION

Abstract:It is not hard to define one-dimensional Brownian motion reflected on an independent one-dimensional Brownian path. I will discuss some path properties of the reflected process and the joint distribution of the two processes. If time permits, I will present a connection with the heat equation, some proofs and an open problem. This is joint work with David Nualart.

Time:Monday, April 16, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Bruce Erickson (University of Washington)

Title:RENEWAL AND OTHER LIMITS FOR SOME SEQUENCES OF RANDOM AFFINE MAPS (PART II)

Abstract:Markov chains defined by random iteration of independent identically distributed maps acting on some space has always been a popular object of study. Of particular interest are chains $\{X_n\}$ of the form: $X_0\in R^d$, $X_n = A_nX_{n-1} + B_n$ where $\{(A_n,B_n)\}$ are independent and identically distributed, with values in $M(d)\times R^d$, $M(d)$ the square matrices of size $d$. An even more popular subset of these objects is the case that each $A_n$ is a random postive multiple of the identity. I will discuss a few more or less recent results, including some rather intricate, but quite interesting ones by Babillot, Bougerol, \& Elie dealing with the asymptotics of the potential kernel (or, if you prefer, the translates of the renewal measure) for the joint chain: $(X_n, \Pi_n)$, $\Pi_n = A_nA_{n-1} \cdots A_1 a$.

Time:Monday, April 9, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Federico Marchetti (Politecnico di Milano)

Title:SPLITTING FREE LUNCHES: FINANCIAL MODELS WITH BOUNDARY CONDITIONS

Abstract:Dirichlet boundary conditions are easily handled in financial models (e.g. in down-and-out barrier options). Boundary reflections are not as acceptable, since they prevent the existence of an equivalent martingale measure and thus lead to arbitrage. We look at a few examples where they might nonetheless be used and suggest how a "fair price" interval (much as in an incomplete market) could be introduced. It should also be noted that the arbitrage opportunities in these models are illusory, since they disappear as soon as transaction costs are included.

Time:Monday, April 2, 2001 at 2:30 p.m.

Location:MLR 302A

Speaker:Bruce Erickson (University of Washington)

Title:RENEWAL AND OTHER LIMITS FOR SOME SEQUENCES OF RANDOM AFFINE MAPS

Abstract:Markov chains defined by random iteration of independent identically distributed maps acting on some space has always been a popular object of study. Of particular interest are chains $\{X_n\}$ of the form: $X_0\in R^d$, $X_n = A_nX_{n-1} + B_n$ where $\{(A_n,B_n)\}$ are independent and identically distributed, with values in $M(d)\times R^d$, $M(d)$ the square matrices of size $d$. An even more popular subset of these objects is the case that each $A_n$ is a random postive multiple of the identity. I will discuss a few more or less recent results, including some rather intricate, but quite interesting ones by Babillot, Bougerol, \& Elie dealing with the asymptotics of the potential kernel (or, if you prefer, the translates of the renewal measure) for the joint chain: $(X_n, \Pi_n)$, $\Pi_n = A_nA_{n-1} \cdots A_1 a$.

Time:Monday, March 5, 2001 at 2:30 p.m.

Location:SMI 111

Speaker:Omer Angel (Microsoft Research)

Title:Random Triangulations

Abstract:Planar triangulations have been studied for separate reasons by mathematiciens (Tutte counted them in 62) and physicists (in the context of quantum gravity). I prove that the uniform measure on random triangulations of the plane with $n$ vertices converge as $n \to \infty$ to some limit distribution of infinite planar triangulations. Some geometric properties of the infinite object are discussed and it is shown to be recurrent.

Time:Monday, February 26, 2001 at 2:30 p.m.

Location:SMI 111

Speaker:Jon A. Wellner (UW)

Title:Some functionals of Brownian motion connected with estimation of montone and convex functions

Abstract:Estimation and testing problems for monotone functions in ``Gaussian white noise'' lead to several interesting functions of two-sided Brownian motion $W$ plus a parabola: the slope process of the greatest convex minorant is now well-understood, thanks to the work of Groeneboom (1983), (1989). In particular, the distribution of the slope process at $0$, say $Z_0$, has been computed analytically and numerically in Groeneboom (1985) and Groeneboom and Wellner (2001). In recent work with Moulinath Banerjee, we have found that the analogue of a chi-square distribution in regular problems is played by the distribution of $$ Z_1 \equiv \int \{ S (t)^2 - S^0 (t)^2 \} dt $$ where $S $ is the slope process of the greatest convex minorant of $W(t) +t^2$ and $S^0$ is the slope process of the one-sided greatest convex minorants constrained to be greater than or equal to zero to the right of zero, and constrained to be less than or equal to zero to the left of zero. An analytical description of the distribution of $Z_1$ is still unknown.

For estimation of a convex function in Gaussian white noise, the maximum likelihood estimator turns out to be the second derivative of a certain ``invelope'' of integrated (two-sided) Brownian motion plus $t^4$. The value $Z_2$ of this second derivative at zero describes the limiting distribution in statistical problems of interest. Almost nothing is known about the distribution of $Z_2$.

I will discuss some of the statistical background for these problems and some of the many open questions.

Time:Friday, February 16, 2001 at 2:30 p.m.

Location:SMI 107

Speaker:Jim Fill (The Johns Hopkins University)

Title:PERFECT SAMPLING: A TALE OF TWO ALGORITHMS

Abstract:In Markov chain Monte Carlo (MCMC), one samples approximately from a probability distribution $\pi$ of interest by (1) designing an ergodic Markov chain X whose stationary distribution is $\pi$ and (2) using X(t), with t suitably large, as an observation approximately from $\pi$. But in practice it is often impossible to determine how large is "suitable" for t.

In the last six years, widely (though not universally) applicable algorithms have been devised that overcome this problem by using the basic mechanisms of MCMC to sampleperfectlyfrom $\pi$. The two most widely applied arecoupling from the past, due to Jim Propp and David Wilson, and an algorithm (calledFMMR) based on rejection sampling, developed by the speaker and extended with coworkers.

I will review these two algorithms, discuss their original (monotone) settings, and explain how the use of so-calledbounding processesanddominating processesextends their ranges of applicability. Along the way, we shall see that there is a simple, previously unnoted, connection between the two algorithms. If time permits, I either will discuss the application of perfect simulation to a distribution arising in the probabilistic analysis of the algorithm "Quickselect" for finding order statistics in an unordered file, or else will discuss how judicious choice of a certain user-supplied parameter (namely, the initial state) can lead to (sometimes dramatic) speedup of FMMR.

(This talk will be based on joint work with Moto Machida, Duncan Murdoch, and Jeff Rosenthal, and on joint work with Bob Dobrow.)

On Monday February 19, 2001, Jim Fill will give a talk at Microsoft Research, titled `` THE RANDOMNESS RECYCLER: A NEW TECHNIQUE FOR PERFECT SAMPLING". The following is the abstract of that talk.

For many probability distributions of interest, it is quite difficult to obtain samples efficiently. Often, Markov chains are employed to obtain approximately random samples from these distributions. The primary drawback to traditional Markov chain methods is that the mixing time of the chain is usually unknown, which makes it impossible to determine how close the output samples are to having the target distribution. Here we present a new protocol, the randomness recycler (RR), that overcomes this difficulty. Unlike classical Markov chain approaches, an RR-based algorithm creates samples drawn exactly from the desired distribution. Other perfect sampling methods such as coupling from the past use existing Markov chains, but RR does not use the traditional Markov chain at all. While by no means universally useful, RR does apply to a wide variety of problems. In restricted instances of certain problems, it gives expected linear time algorithms for generating samples. I will discuss how RR applies to self-organizing lists, the Ising model, random independent sets, random colorings, and the random cluster model. (This talk will be based on joint work with Mark Huber.)

Time:Monday, February 12, 2001 at 2:30 p.m.

Location:BAG 331A

Speaker:Laszlo Lovasz (Microsoft Research)

Title:A refined conductance bound on mixing times and sampling from convex bodies

Abstract:To get sharp bounds on the mixing times of a large variety of finite Markov chains is an important issue for many applications in computer science, physics, etc. One very successful technique to prove rapid mixing of Markov chains, introduced by Jerrum and Sinclair, is the use of "conductance". However, the bound on the mixing time in terms of the conductance, involves also the logarithm of the smallest stationary probability. This factor, sometimes called the "start penalty", is often an artifact of the proof: if we know better bounds on the conductance of small sets, than we can reduce it. Several techniques have been proposed to do so, most notably the "log-Sobolev" inequalities by Diaconis and Saloff-Coste. We prove a new inequality that bounds the mixing time in terms of a weighted average, where conductances of small sets get larger weights. We show various applications of this new inequality; in particular, one gets a new bound on the mixing time of the random walk in a d-dimensional convex body (with ball steps), which is optimal in terms of the diameter. This requires a new isoperimetric inequality for small subsets of a convex sets. Joint work with Ravi Kannan from Yale.

Time:Monday, February 5, 2001 at 2:30 p.m.

Location:SMI 111

Speaker:Federico Marchetti (UW and Politecnico di Milano)

Title:QUEUES WITH COMPETING ARRIVALS (II)

Abstract:Diffusion limits for queuing systems in "heavy traffic" promise to provide tractable models for important questions in queuing applications where different arrival streams compete for limited resources. We will illustrate some early results and some suggestions for further research. The models that arise involve mainly reflecting diffusions, both as descriptive models and as tools for possible optimal control questions.

Time:Monday, January 22, 2001 at 2:30 p.m.

Location:SMI 111

Speaker:Zhen-Qing Chen (UW)

Title:Conditional Gauge Theorem

Abstract:When $X$ is a Brownian motion and $D$ is a bounded smooth domain, it is known that under certain condition on potential $q$, function $u(x, y)=E_x^y[ \exp(\int_0^{\tau_D} q(X_s)ds)]$ is either identically infinity or is bounded on $D\times D$. This result is called conditional Gauge theorem. In this talk we will show that above theorem holds for a large class of processes, including processes with jumps. The significance of the conditional gauge function $u$ is that it is the ratio of the Green function for Schrodinger operator $L+q$ in $D$ and the Green function for $L$ in $D$, where $L$ is the generator of process $X$.

Time:Monday, January 8, 2001 at 2:30 p.m.

Location:SMI 111

Speaker:Chris Burdzy (UW)

Title:The supremum of Brownian local times on H\"older curves

Abstract:For $f: [0,1]\to \R$, let $L^f_t$ be the local time of space-time Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions whose H\"older norm of order $\al$ is less than or equal to 1. We show that the supremum of $L^f_1$ over $f$ in $\sS_\al$ is finite if $\al>\frac12$ and infinite if $\al<\frac12$.

Joint work with R.~Bass.

Time:Monday, November 20, 2000 at 2:30 p.m.

Location:EEB 216

Speaker:Marek Biskup (Microsoft Research)

Title:LONG-TIME TAILS FOR DIFFUSIONS IN RANDOM MEDIA: PARABOLIC ANDERSON MODEL WITH BOUNDED POTENTIALS

Abstract:I will describe the long-time behavior of the solution to a parabolic second-order differential problem with a random i.i.d. potential. In the literature, this problem appears under the name "parabolic Anderson model," referring to the Anderson Hamiltonian which is widely studied in the context of disordered quantum systems. I will show that the leading order asymptotic of the moments of the solution as well as the solution itself can be described in terms of variational principles. As an application, the Lifshitz tails for the spectrum of the associated Schroedinger operator with random i.i.d. potential can be explicitly computed. The Lifshitz exponent varies between half spatial dimension and infinity, depending on the upper tail of the potential distribution. The main tools of the proof are the Feynman-Kac formula and estimates for principal eigenvalues. The results extend various findings about the "simple random walk among Poissonian obstacles" obtained by Sznitman and his school in the 1990s.

Jointly with Statistics

Time:Monday, November 13, 2000 at 3:30 p.m.

Location:COM 120

Speaker:Krzysztof Burdzy (University of Washington)

Title:REMARKS ON THE PHILOSOPHY OF BAYESIAN DECISION MAKING

Abstract:I will present a thought experiment showing that a Bayesian decision maker is forced in some circumstances to identify himself with an abstract group of people (as opposed to a real group). The analysis showing sub-optimality of the Bayesian decision will be made from the frequentist point of view. If time permits, I will discuss an unstated axiom in Bayesian textbook examples and its practical applications. The axiom is far from obvious and thus destroys the claim of the Bayesian approach to be based on an axiomatic system.

The talk will be partly based on the notes available at http://www.math.washington.edu/~burdzy/Bayes/welcome.html .

Time:Monday, November 6, 2000 at 2:30 p.m.

Location:EEB 216

Speaker:Elchanan Mossel (Microsoft Research)

Title:GLAUBER DYNAMICS AND THE ISING MODEL ON THE TREE

Abstract:Glauber dynamics on finite graphs are used in order to sample Markov chains such as the Ising model and random colorings.

We will recall the classical picture of the Ising model on Z^2 where in the uniqueness phase, the spectral gap is bounded away from zero, while in the non-uniqueness phase, the spectral gap decays exponentially in the diameter of squares. In recent years, effort was devoted to generalize this equivalence.

By studying these dynamics on trees, we will show that for general graphs this equivalence doesn't hold. We will show that bounded spectral-gap implies in general a property (*) of the Gibbs measure. Moreover, for trees and Z^2, the spectral-gap is bounded if and only if (*) holds.

Based on a joint work with Claire Kenyon and Yuval Peres.

Time:Monday, October 30, 2000 at 2:30 p.m.

Location:EEB 216

Speaker:Federico Marchetti (Politecnico di Milano)

Title:FINITE-BUFFER QUEUES: ALLOCATING LOSSES AMONG COMPETING INCOMING FLOWS IN HEAVY TRAFFIC

Abstract:Routers and switches provide finite buffers for incoming traffic and packet losses will occur. It does make a difference whose packets get lost, especially because different flows will work under different protocols and react differently to losses. Overall network efficiency might be affected depending on the protocol. Keeping track of packet labels poses no problem (in principle), but the heavy traffic (diffusion) limit of the loss processes is less obvious. We discuss some preliminary analysis of this limit.

Time:Tuesday, October 24, 2000 at 2:30 p.m.

Location:LOW 222

Speaker:Itai Benjamini (Weizmann Institute of Science and Microsoft Research)

Title:GEOMETRIC RANDOM GRAPHS

Abstract:I will present several results regarding the geometric structure of various models of random graphs, and the behavior of random walk on these graphs.

University of Washington Walker-Ames Lecture

Time:Wednesday, October 18, 2000 at 7:00 p.m.

Location:Kane 210

Speaker:Persi Diaconis (Stanford University)

Title:On coincidences

Abstract:Coincidences amaze us. I will review early work by Jung and Freud and also show how sometimes a bit of quantitative thinking can show that things aren't so surprising after all.

A reception will follow the lecture at the lobby.

Departments of Mathematics and Statistics Joint Colloquium

Time:Tuesday, October 17, 2000 at 4:00 p.m.

Location:MLR 301

Speaker:Persi Diaconis (Stanford University)

Title:What do we know about the metropolis algorithm?

Abstract:The metropolis algorithm is one of the most widely used tools of 20th century scientific computing. I will explain the algorithm, illustrate its use in cryptography and chemistry and describe new geometric tools for bounding rates of convergence.

A reception will be held at Faculty club after the colloquium.

Seminar on Mathematical Education

(sponsored by The University of Washington Teaching Academy, The Department of Statistics, The Department of Mathematics, and the Center for Quantitative Science)

Time:Tuesday, October 17, 2000 at 12:30 p.m.

Location:Anderson 22

Speaker:Susan Holmes (Stanford University)

Title:Probability by Surprise: the pleasure of paradoxes

Abstract:Probability by Surprise: Teaching with Paradoxes

The main idea is to unify the presentation of probability to a heterogenous audience through the interest we have in things that surprise us. Some examples we use in our probability classes include: 'the birthday problem', 'say red', 'russian roulette', 'de Mere's problem', 'Monty Hall'.

The tools developed are based on discoveries by cognitive pyschologists (in particular Tversky and Kanneman) over the last 20 years, that have not, as yet, been used in teaching probability in this country. The programs include simulation programs to give students a feel for probability, animated scenarios to help motivate and amuse them, as well as to make the material more memorable.

The project weaves together an "Introduction to Probability'' web-site including: - links to useful existing calculus material such as explanations of integration and summing rules. - interactive Venn diagrams, probability trees, densities as limits of histograms. - Relevant graphical animations written in java. - Historical material about probabilty. - Animations and simulations developed specifically. - Lists of team oriented project ideas that enable the students to try out their new computer simulation skills and compare these results to those of classical probabilistic analyses.

Joint Statistics and Biostatistics Seminar

Time:Monday, October 16, 2000 at 3:30 p.m.

Location:Communications 120

Speaker:Susan Holmes (Stanford University)

Title:Confidence Regions for Phylogenetic trees

Abstract:We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of tree

This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.

New types of confidence statements are made possible in this cube complex.

(Coffee/Tea and Cookies in the Statistics Lounge after the seminar)

Time:Monday, October 16, 2000 at 2:30 p.m.

Location:MLR 301

Speaker:Persi Diaconis (Stanford University)

Title:Probability, statistics and the zeros of the zeta function.

Abstract:Random matrix theory uses the eigenvalue distribution of typical, large matrices to model natural phenomena. I will show that these models give remarkable fits to a large real data set (50,000 zeros of Riemanns zeta function). This opens up new problems in testing goodness of fit with dependent data.

Time:Monday, October 9, 2000 at 2:30 p.m.

Location:EEB 216

Speaker:Lisa Korf (University of Washington)

Title:PRICING CONTRACTS CONTINGENT ON A MARKET: A STOCHASTIC PROGRAMMING PERSPECTIVE

Abstract:The classical fundamental theorem of asset pricing, and its variants, say in essence that a market is arbitrage free if and only if there is an equivalent probability measure with respect to which the market price process is a martingale. The fair price of a financial instrument, i.e. the minimum initial investment in the market that permits replication of the instrument, may then be computed as an expectation of the contracted payouts of the instrument with respect to such a measure. When this problem of pricing contracts contingent on a market is cast in a stochastic programming setting, it opens up the possibility to analyze (and solve) much more complex problems than would be considered in the classical setting. Duality plays a central role. Some of the key features that come out of this approach will be presented.

Time:Monday, October 2, 2000 at 2:30 p.m.

Location:EEB 216

Speaker:Hong Qian (University of Washington)

Title:THE MATHEMATICAL THEORY OF SINGLE MOLECULES AS A MECHANICAL ENGINE

Abstract:A single molecule of a protein can move, along its designated track, against external applied force in aqueous solution. This process is the physical basis of the cellular movement which leads to many aspect of biological motion, for example, muscle contraction and wound healing. We will introduce this fascinating phenomenon and its mathematical model in terms of a set of coupled diffusion process. Some unsolved mathematical issues will be discussed.

Time:Wednesday, May 31, 2000 at 2:30 p.m.

Location:Smith 115

Speaker:S\"onke Lorenz (Freie Universit\"at Berlin and University of Washington)

Title:TRANSFER OPERATOR APPROACH TO CONFORMATIONAL DYNAMICS IN BIOMOLECULAR SYSTEMS

Abstract:In my talk, I am going to present the development of novel mathematical concepts and algorithmic approaches that result in modelling long-term behavior of biomolecular systems by applying conformational dynamics.

Both, a first deterministic approach, based on the Frobenius-Perron operator corresponding to the flow of the Hamiltonian dynamics, and later stochastic approaches, based on a spatial Markoff operator and on Langevin dynamics, can be subsumed under the unified mathematical roof of the transfer operator approach to effective dynamics of molecular systems.

At the end, I plan to illustrate the concept by numerical results.

Joint work with Christof Sch\"utte, Wilhelm Huisinga and Peter Deuflhard from the Freie Universit\"at Berlin und the Konrad-Zuse-Zentrum Berlin (ZIB).

Time:Monday, May 15, 2000 at 2:30 p.m.

Location:Smith 115

Speaker:Bruce Erickson (University of Washington)

Title:A LIMIT THEOREM FOR IMBEDDED REGENERATIVE SETS (Continued)

Abstract:See previous one

Mathematics Department Colloquium

Time:Tuesday, May 23, 2000 at 4:00 p.m.

Location:PDL C-36

Speaker:Michael Cranston (University of Rochester)

Title:Dispersion of stochastic flows

Abstract:A problem in statistical fluid mechanics is to determine how fast a pollutant (passive tracer) will spread. We will discuss some models of motion of so-called 'passive tracers' and give some results on how far they might travel in a given amount of time. In one model each passive tracer undergoes a Brownian motion and the Brownian motions experienced by different tracers will have a given correlation. In another model the tracers move according to a random velocity field. This is the model given by Kolmogorov. The results in each case are similar and maybe a little surprising.

Time:Monday, May 22, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Katalin Marton (R\'enyi Institute, Hungary)

Title:The "transportation cost" method to prove measure concentration

Abstract:The concentration of measure phenomenon, in product spaces, means the following: If a subset $A$ of the $n$'th power of a probability space $\Cal X$ has not too small probability then very large probability is concentrated on a small neighborhood of $A$. The neighborhood is, understood, e.g., with respect to Hamming distance.

A few years ago, M. Talagrand's work put measure concentration in the focus of attention of many people, proving a variety of interesting inequalities and showing lots of applications.

In this talk we show how to prove measure concentration by bounding some distance between probability measures, (e.g., $\bar d$-distance) in terms of the relative entropy of these measures. This method can be used to prove measure concentration for dependent random variables (or random fields) under the assumption that the given ensemble of random variables satisfies a strong mixing condition, in the sense of Dobrushin and Shlosman.

Mathematics Department Colloquium

Time:Tuesday, May 9, 2000 at 4:00 p.m.

Location:Thomson 125

Speaker:R. J. Williams (University of California, San Diego)

Title:Dynamic control of stochastic networks in heavy traffic

Abstract:Stochastic networks are used as models for complex manufacturing, telecommunications and computer systems. Some of these networks allow for flexible scheduling of jobs through dynamic (state-dependent) alternate routing and sequencing, hereafter collectively referred to as dynamic scheduling. Usually these models cannot be analyzed exactly, and it is a challenging problem to design dynamic scheduling policies for such networks that are simple to implement and yet are approximately optimal in an appropriate sense. As one approach to this problem, J. M. Harrison proposed the use of Brownian control problems (BCPs) as formal heavy traffic approximations to dynamic scheduling problems for stochastic networks. This talk will survey developments in the application and justification of this approach.

Time:Monday, May 8, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Bruce Erickson (University of Washington)

Title:A LIMIT THEOREM FOR IMBEDDED REGENERATIVE SETS

Abstract:If ${\bf R}\subset [0,\infty)$ denotes a closed unbounded random set of points and $Z_t(\omega) = 1_{{\bf R}(\omega)}(t)$ is the process of indicators, then ${\bf R}$ is a {\it regenerative set} if for any ${\bf F}_t\equiv\sigma\{Z_s, s\le t\}$--stopping time $T$ with $P\{T\in {\bf R}\} = 1$, the subset ${\bf R}\circ\!\theta_T = \{ s : s +T\in {\bf R}\}\cap [0,\infty)$ is independent of ${\bf F}_T$ and has the same law as ${\bf R}$. When the regenerative set is discrete, there exists a unique renewal process $S = \{S_n\}_{n\in N}$ such that ${\bf R} = S$. (A renewal proces is a sequence of partial sums of non-negative i. i. d. random variables: $S_k = \xi_1 + \cdots + \xi_k$.)

The familiy of random variables $$ A_t = \inf\{ s \ge 0 : t-s \in {\bf R} \}, \quad t \ge 0, $$ is called the {\it age process} (or ``spent waiting time at epoch $t$'').

In the discrete, non-lattice case, if the renewal jumps have finite mean, $m$, then the Markov process $A_t$ has a limit distribution (for $t \to \infty$) and the limit distribution has density: $\,m^{-1}P\{\xi > x\}\,\hbox{d}x$. This is a consequence of the strong renewal theorem.

If $m=\infty$, then $A_t/t$ has a non-trivial limit distribution if and only if $\xi$ belongs to the domain of attraction of a positive stable law. This result is due to Lamperti and, independently, Dynkin. The limit is a ``beta'' type distribution. (Note that the age process at time $t$ is bounded by $t$ and this fact combined with certain properties of regular varying functions show that the Dynkin-Lamperti result has the equivalent formulation: $A_t/t$ converges in distribution iff $E\{A_t\}/t$ has a limit in $(0,1)$.)

I will discuss a generalization, due to J. Bertoin, of the Lamperti-Dynkin result in the case of a finite sequence of embedded discrete regenerative sets. Naturally, I will have to tell you the meaning of the latte

Time:Monday, April 24, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Krzysztof Bogdan (Technical University of Wroclaw and UW)

Title:ON SCHR\"ODINGER TYPE OPERATORS BASED ON FRACTIONAL LAPLACIAN (continued)

Abstract:We will discuss weakly and strongly harmonic functions for Feynman-Kac semigroups corresponding to symmetric stable processes. Based on the properties of the weak Schr\"odinger operator we will then give examples of the gauge function for bounded Green domains. Larger domains, like half-lines, will be also mentioned in the context of the Kelvin transform.

Time:Monday, April 17, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Wenbo Li (University of Delaware)

Title:CAPTURE TIME OF BROWNIAN PURSUITS

Abstract:Let $B_0, B_1, \cdots, B_n$ be independent standard Brownian motions, starting at $0$. Define the stopping time $$ \tau_n = \inf \{ t > 0: B_i (t) -1 = B_0 (t) \; for some \; 1 \le i\le n \}. $$ A proof of the fact that $E \tau_3 =\infty$ and $E \tau_5 < \infty$ will be presented, along with various related problems.

Time:Monday, April 10, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Krzysztof Bogdan (Technical University of Wroclaw and UW)

Title:ON SCHR\"ODINGER TYPE OPERATORS BASED ON FRACTIONAL LAPLACIAN

Abstract:We will discuss some elements of the potential theory of Feynman-Kac semigroups corresponding to symmetric stable processes. We will focus on problems related to gaugeability and existence of $q$-harmonic functions.

Time:Monday, April 3, 2000 at 2:30 p.m.

Location:MEB 250

Speaker:Ron Pyke (University of Washington)

Title:ON RANDOM WALKS AND DIFFUSIONS RELATED TO PARRONDO'S GAMES

Abstract:Consider playing a seguence of simple games, with $X_n$ denoting the gain (or loss) from the $n$-th game. If $S_n = X_1+\cdot \cdot \cdot +X_n$ is the player's capital after $n$ games, we say that the game is winning/losing/fair according as $S_n/n$ converges $a.s.$ to a positive/negative/zero limit. In a recent series of papers, G. Harmer and D. Abbott study the behavior of random walks associated with games introduced in 1997 by J. M. R. Parrondo. These games illustrate an apparent paradox that random and deterministic mixtures of losing games may produce winning games. In this talk, classical cyclic random walks on the additive group of integers modulo $m$, a given integer, are used in a straightforward way to derive the strong law limits of a general class of games that contains the Parrondo games. We then consider the question of when random mixtures of fair games related to these walks may result in winning games. Although the context for these problems is elementary, there remain open questions. An extension of the structure of these walks to a class of shift diffusions is also presented, leading to the fact that a random mixture of two fair shift diffusions may be transient to $+\infty$.

Time:Tuesday, March 14, 2000 at 2:30 pm.

Location:PDL C-36

Speaker:Tadeusz Kulczycki (Technical University of Wroc\l aw)

Title:Intrinsic ultracontractivity for symmetric stable processes

Abstract:I will talk about intrinsic ultracontractivity (IU) for the semigroup $P_{t}^{D}$ generated by the symmetric $\alpha$-stable process "killed" on exiting an open set $D$ with finite Lebesgue measure. Intrinsic ultracontractivity has been extensively studied in the case of the Brownian motion. IU gives pointwise estimates of consecutive eigenfunctions by the first one and is equivalent to parabolic boundary Harnack principle.

The main result states that if $D$ is a bounded open set then $P_{t}^{D}$ is IU. I am also going to talk about IU for some unbounded open sets D.

Time:Monday, March 7, 2000

Location:Raitt 109

Speaker:Bert Schreiber (U of Washington and Wayne State U.)

Title:From Stationary to Nonstationary Processes on Groups: How Can We Extend the Spectral Analysis?

Abstract:The classical theory of stationary random processes and fields is founded upon the fact that the covariance function of such a field has an associated spectral measure. The talk will focus on several approaches to dealing with classes of nonstationary processes and fields. The classes we will discuss all have elements with some sort of spectral decomposition and all contain the stationary fields. First we will recall two types of harmonizable processes. The first class is defined by a direct extension of the spectral representation for stationary processes. The second type of harmonizable process involves a similar but much broader notion of spectral representation, which we will describe.

Next we will introduce a class of processes and fields called asymptotically stationary. For many processes that arise in the real world, it is the discrete spectrum that is of primary interest, the continuous spectrum being attributable to "noise." We will describe a recent result of L. Hanin and the speaker which provides a consistent statistical estimator for the coefficients of the discrete part of the associated spectral measure.

Time:Monday, February 28, 2000

Location:Raitt 109

Speaker:Minping Qian (University of Southern California and Beijing University)

Title:The Metastability Behavior of Markov Chains and Diffusion--A Large Deviation Consideration

Abstract:For a Markov chain, suppose it has transition probability prooportional to $ exp(-bH(x)) $ for very large $b$. As $b$ goes to infinite, this process may not be irreducible, its state space can divided into several parts, each part has its recurrent core (we say attractor). The limit may be degenerated to deterministic processes, if we restrict it in recurrent cores. These subspace of recurrent cores , are metastability sets (or points).There is a hierarchical structure in these subspaces. The path of original processes has a kind of special behavior moving in this hierarchical structure. This understanding has important application in Markov Monte Carlo, simulated annealing and other stochastic algorithms. This kind of structure can also be found for diffusions and nonsymmetric Markov processes.

Time:Wednesday, February 23, 2000

Location:THO 211

Speaker:Tusheng Zhang (UC Irvine and HIA in Norway)

Title:On the Small Time Large Deviations of Diffusion Processes on Hilbert Spaces

Abstract:In this talk, I will present results on the following small time large deviations of Varadhan type $$\lim_{t\rightarrow 0} 2t log P(X_0\in A, X_t\in B)=-d^2(A,B)$$ for diffusions on Hilbert spaces, which particularly includes solutions of SPDEs.

Time:Monday, February 14, 2000 at 2:30 pm.

Location:Raitt 109

Speaker:Bruce Erickson (University of Washington)

Title:The collection of recurrent (or transient) subsets for normal walks on Z^d (d >2) is independent of the walk

Abstract:This will be a continuation of my talk on February 7. A normal random walk on Z^d is one with an aperiodic step distribution which has mean (vector) 0 and finite variance.

Time:Monday, February 7, 2000 at 2:30 pm.

Location:Raitt 109

Speaker:Bruce Erickson (University of Washington)

Title:Some Recent Results in the Potential Theory of Normal Random Walks in Z^d.

Abstract:A normal random walk is one whose step distribution is aperiodic and has mean (vector) 0, and whose step length has a finite 2nd moment. This talk will be a brief report on some recent ('96 to '98) results due, mainly, to K. Uchiyama on the potentials of such walks. What is (mildly) surprising is that some of the consequences of this work had not already been established 35 years ago when this sort of thing was getting much greater attention.

By the way, the term ``normal'' is not normal. But it does make titles to talks a little more normal in size. Moreover, a normal random walk is in the domain of normal attraction of a normal distribution!

Time:Monday, January 31, 2000 at 2:30 pm.

Location:Raitt 109

Speaker:Christopher Hoffman (University of Washington)

Title:Simple Random Walk on Percolation Clusters

Abstract:Consider an infinite percolation cluster in Z^d. We perform simple random walk on this graph. We obtain bounds on the probability that simple random walk started at vertex at x at time 0 returns to x at time t. This is a joint work with Debby Heicklen.

Time:Tuesday, January 25, 2000 at 2:30 pm.

Location:RAI 116

Speaker:Michiel van den Berg (University of Bristol, UK)

Title:Heat Flow, Area and Capacity for Regions with Many Small Holes

Abstract:We consider the heat flow from a compact set K in euclidean space R^m if K is kept at fixed temperature 1 while R^m - K has initial temperature 0. We obtain estimates for the total heat content in R^m - K at time t as t goes to 0 in the special case where K is an infinite union of closed non-intersecting balls.

Time:Wednesday, January 19, 2000 at 2:30 pm.

Location:CMU 230

Speaker:Rick Durrett (Cornell University)

Title:Single Nucleotide Polymorphims in the Human Genome: How many are there? How many will Celera find?

Abstract:Single nucleotide polymorphisms (SNP's) are useful markers for locating genes since they occur throughout the human genome and thousands can be scored at once using DNA microarrays. Here, we use branching processes and coalescent theory to show that if one assumes a mutation rate of 5 x 10^{-9} per nucleotide per generation then there are 2.8 million SNP's in the human genome, or one every 1,063 base pairs. We further predict that Celera's strategy for sequencing the human genome will uncover 1.8 million variable nucleotides but that in only about 72% of these cases will the most common allele have a frequency of less than 99%. This means, however, that they will find approximately 45% of the SNP's in the human genome. The predicted numbers are of course proportional to the assumed mutation rates but the percentage estimates are independent of it.

Time:Monday, January 10, 2000 at 2:30 pm.

Location:RAI 109

Speaker:Zhen-Qing Chen (University of Washington)

Title:Strong Solutions of Stochastic Differential Equations for Dirichlet Processes and Pathwise Uniqueness

Abstract:Strong solution and pathwise uniqueness for stochastic differential equations (SDE) related to second order non-divengence form differential operator is well studied. It is an open question whether there is a strong solution for "SDE" related to second order divergence form differential operator with non-Lipschitz coefficients. In this talk, we will present some recent results on the existence of strong solution and pathwise uniqueness for one dimensional

1) SDE where the drift term is a signed measure,

2) SDE related to divergence form operator,

3) Stratonovitch SDE,

as well as some non-existence and non pathwise uniqueness results.

This talk is based on a joint research project with Rich Bass.

Time:Monday, November 22, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Krzysztof Burdzy (University of Washington)

Title:SONIC BOOM FOR THE HEAT EQUATION

Abstract:Encyclopaedia Britannica has this to say about ``sonic boom,'' among other things: ``As the aircraft proceeds, the trailing parabolic edge of that cone of disturbance intercepts the earth, producing on earth a sound of a sharp bang or boom--with silence before and after.'' I will show that the heat equation may display a similar behavior in a time-variable domain. This is the fourth talk in a series on the heat equation and reflected Brownian motion in time-dependent domains. The previous three talks were given by the co-authors---Zhenqing Chen and John Sylvester.

Time:Monday, November 15, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Federico Marchetti (Politecnico di Milano)

Title:EVALUATING STOCHASTIC ALGORITHMS

Abstract:For a convergent stochastic algorithm, the last entrance time into a prescribed neighborhood of the limit, seems a more meaningful parameter than, for instance, the first entrance time. To illustrate this point, we look at scaling properties and other features that are of interest in this application, for some simple cases, and suggest further lines of investigation.

Joint work with P. Glynn, Stanford.

Time:Monday, November 8, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Yimin Xiao (Theory Group, Microsoft)

Title:LEVEL SETS OF ADDITIVE LEVY PROCESSES

Abstract:Let $X = \{ X(t); t \in {\bf R}^N_+ \}$ be an additive L\'evy process in ${\bf R}^d$. We show that the hitting probabilities of the level sets of $X$ are related to a class of natural capacities on ${\bf R}^N_+$.

We also present several probabilistic applications of the aforementioned potential--theoretic connections. They include areas such as intersections of L\'evy processes and level sets, as well as Hausdorff dimension computations.

Joint work with Davar Khoshnevisan.

Time:Monday, November 1, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Zhenqing Chen (University of Washington)

Title:REFLECTING BROWNIAN MOTION IN TIME-VARYING DOMAIN AND ITS ASSOCIATED PDE (PART II)

Abstract:In this talk, I will first review reflecting Brownian motion in a time-varying domain that I discussed in the first talk. Then I will present its various properties and connections with PDE's, including various probabilistic representation for solutions to heat equation with a moving insulated boundary. Both one and higher dimensional cases will be considered. This is a part of a joint research project with Chris Burdzy and John Sylvester.

Time:Monday, October 25, 1999 at 2:30 p.m.

Location:THO 231

Speaker:K.B. Erickson (University of Washington)

Title:SOME PROBLEMS IN SINGULAR BRANCHING BROWNIAN MOTION

Abstract:A Brownian particle with initial position $\,a\,$ moves for a random time $\tau$ with conditional distribution depending on the path. At this time the particle splits in two. The new particles, given their initial position, $X_\tau$, are mutually independent and are governed by the same distributional laws as their parent. Etc. As time goes by, we get an increasing number of not quite independent Brownian particles. If $\#(t;I)$ denotes the the number of particles in $\,I\,$ at time $t$ and if the death rate has a particular form, one finds that $\#(t;I)< \infty$ for all $t$ with probability 1, but that $E^a \#(t;I) = \infty\, $ for all initial positions $a$ and all $t$ sufficiently large. One interesting question concerns the rate of growth of the spread of the process. Specifically, if $z(t)$ is the position of the right-most particle at time $t$, $z(t)=\max\{ b:\#(\, t; \, \ico{b}{\text{$\infty$}}\,) \neq 0 \} $, and $z_+(t) = \max(\, z(s);\, s \le t)$, then $z_+(t) \uparrow \infty$, but what is the rate of increase of $z_+(t)$ ?

(Joint with COMPLEX ANALYSIS SEMINAR)

Time:October 19, 1999 at 1:30 p.m.

Location:PDL C-401

Speaker:Balint Virag (University of California, Berkeley)

Title:BROWNIAN BEADS

Abstract:Two-dimensional stochastic models are of great interest to physicists and probabilists because they reflect, perhaps more than any other mathematical object, the rich world of conformal symmetries.

We consider perhaps the simplest one of these models, 2D Brownian motion. Two and three are the only dimensions where Brownian motion has interesting cut-times, that is times at which past and future paths are disjoint. Brownian beads are the sections of the path in between the cut-times. We show that these beads are in some sense independent of each other and 2D Brownian motion is stringed out of them just as 1D Brownian motion is built out of excursions.

As an application, we give a heuristic explanation for some conjectured formulas of Duplantier, Lawler and Werner about intersection exponents for planar Brownian motion.

Time:Monday, October 18,, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Zhenqing Chen (University of Washington)

Title:REFLECTING BROWNIAN MOTION IN TIME-VARYING DOMAIN AND ITS ASSOCIATED PDE

Abstract:In this talk, I will describe a diffusion process associated with the heat equation with a moving insulated boundary which John Sylvester discussed in our last probability seminar. This process is a reflecting Brownian motion in a time-varying domain. I will present results on existence and uniqueness of such processes. I will also discuss their various properties and connections with PDE's. Both one and higher dimensional cases will be considered. This is a part of a joint research project with Chris Burdzy and John Sylvester.

Time:Wednesday, October 13, 1999 at 2:30 p.m.

Location:THO 134

Speaker:John Sylvester (University of Washington)

Title:A HEAT EQUATION WITH A MOVING INSULATED BOUNDARY

Abstract:In recent conversations with Chris Burdzy and Zhen-Qing Chen, we have found that analysts and probabilists ( at least this analyst and those probabilists) have quite different perspectives on the process of diffusion. I will present an elementary description, from an analysts point of view, of how one formulates the boundary value problem alluded to in the title, and the estimates one must obtain to prove existence and uniqueness.

Time:Monday, October 4, 1999 at 2:30 p.m.

Location:THO 231

Speaker:Krzysztof Burdzy (University of Washington)

Title:SYNCHRONOUS COUPLINGS IN NON-CONVEX DOMAINS

Abstract:Consider two Brownian particles trapped in a domain, i.e., consider two reflected Brownian motions. Suppose that they move in a synchronous way if both are away from the boundary. It is easy to see that the distance between the particles will go to $0$ if the domain is convex. Is this also true in non-convex domains? (Joint work with Z.-Q. Chen.)

Time:Wednesday, Septemeber 29, 1999 at 2:30 p.m.

Location:SMI 113

Speaker:Michal Karonski (Adam Mickiewicz University, Poznan, Poland)

Title:PHASE TRANSITION IN RANDOM GRAPHS AND HYPERGRAPHS

Abstract:The talk gives an overview of the basic ``epochs'' in the evolution of random graphs and hypergraphs. Special attention is given to the phase transition phenomenon. In particular, we present recent joint results with Tomasz Luczak on the phase transition in a random $d$-uniform hypergraph and the size of its largest component in the sub- and supercritical phases.

Time:Thursday, August 19, 1999 at 2:30 p.m.

Location:RAI 109

Speaker:Yuval Peres (University of California, Berkeley)

Title:PERCOLATION IN A DEPENDENT RANDOM ENVIRONMENT

Abstract:Draw planes in $R^3$ that are orthogonal to the $z$ axis, and intersect that axis at the points of a Poisson process with intensity $\lambda$; similarly, draw planes orthogonal to the $x$ and $y$ axes using independent Poisson processes (with the same intensity) on these axes. This yields a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length $\ell$ is (independently) open with probability $e^{-\ell}$. We show that this model exhibits a phase transition: For large enough $\lambda$, there is an infinite open cluster a.s., and for small $\lambda$, all open clusters are finite a.s. (The question whether the analogous process in two dimensions exhibits a phase transition is open.) We prove this result using the method of `Paths With Exponential Intersection Tails'; This method yields a proof of phase transition for a large class of percolation processes in random environment in dimension $d \geq 3$.

Joint work with Johan Jonasson and Elchanan Mossel.

Time:Wednesday, August 11, 1999 at 2:30 p.m.

Location:Smith 109

Speaker:Itai Benjamini (Weizmann Institute)

Title:GEOMETRIC EMBEDDINGS AND PROBABILITY

Abstract:We will present several loosely related results regarding coarse geometric embeddings, mostly of graphs. We apply probabilistic, geometric and potential-theoretic arguments. Joint work with Oded Schramm.

Time:Tuesday, July 27, 1999 at 2:30 p.m.

Location:Smith 111

Speaker:Siva Athreya (University of British Columbia)

Title:THE DUALITY METHOD FOR VARIOUS MARTINGALE PROBLEMS

Abstract:Duality is a useful tool in showing uniqueness and other properties for solutions of martingale problems. We shall begin with a basic review of the method and then move onto applications with special emphasis on uniqueness issues for solutions of certain Stochastic differential equations and SPDE's.

This is joint work with Roger Tribe.

Time:Wednesday, June 2, 1999 at 2:30 p.m.

Location:Smith 311

Speaker:Paul Shields

Title:Why should a probabilist care about ergodic theory?

Abstract:One result from ergodic theory sometimes used, implicitly or explicitly, by probabilists is the ergodic theorem. Many also know about the Ornstein isomorphism theory, but as far as I know no probabilist has made use of it. I would like to discuss a recent application of isomorphism theory to an approximate-match waiting-time question, a question that probabilists might find of interest or even use someday. (In fact, some of what I discuss can be viewed as a generalized form of Talagrand's "new view of independence.")

Time:Wednesday, May 26, 1999 at 3:30 p.m.

Location:Smith 107

Speaker:ROBERT ADLER (Technion)

Title:SOME INTERACTING SUPERPROCESSES

Abstract:The talk will focus on some new perturbations of basic superprocess, generally involving highly singular self-interactions.

Abstract for the non-initiate (for whom the talk is really intended): The talk will be composed of four parts;

1: A brief introduction to superprocesses at the level of systems of branching, moving, particles. (For which you have to know only what Brownian motion is.)

2: An explanation of 2-3 ways of introducing smooth interactions (between particles) into the system. (For which you have to either watch my hands or read stochastic PDE's.)

3: A divergence into how to look at highly local interactions between two Brownian motions, and an explanation of what this has to do with Burgers' equation. (Including an explanation of what this equation is and why it is interesting.)

4: A discussion of a superprocess based on the above non-linear interaction, which turns out to be the same as looking at Burgers' equation with a random forcing term, and why this is interesting.

Time:Tuesday, May 25, 1999 at 3:30 p.m.

Location:Smith 205

Speaker:ROBERT ADLER (Technion)

Title:TWENTY FIVE YEARS LATER

Abstract:In 1973 and 1974 Ron Pyke published two review papers, one entitled "Partial sums of matrix arrays, and Brownian sheets" and the other "Multidimensional empirical processes: some comments". Both not only surveyed the literature of the time, but also posed a number of problems, which, with hindsight, set the scene for frenetic research activity that has continued to this day.

While it would be impossible to summarise all of this activity in one hour, or even to cover Ron's own contributions to the area, I will describe a number of the developments that would seem to be of widest interest. In particular, I will present some rather enticing problems in the random geometry generated by multi-parameter stochastic processes of both theoretical and applied interest.

(Remark: The talk is a part of Ron Pyke's retirement celebration.)

Time:Monday, May 10, 1999 at 2:30 p.m.

Location:LOW 216

Speaker:Kathy Temple (University of Washington)

Title:DIFFUSION-LIMITED AGGREGATION

Abstract:Diffusion-Limited Aggregation, or DLA, is a very simple process to describe but extraordinarily difficult to analyze. This talk will present the model and review a few known results about DLA.

Time:Monday, April 26, 1999 at 2:30 p.m.

Location:LOW 216

Speaker:Rajesh Nandy (University of Washington)

Title:A BRIEF REVIEW OF QUANTUM STOCHASTIC CALCULUS (continued)

Abstract:The basic integrator processes of Quantum Stochastic Calculus are introduced in appropriate Hilbert Space. Also the Quantum Ito Formula is described. Relevant examples will be given as well.

Time:Monday, April 19, 1999 at 2:00 p.m.

Location:LOW 216

Speaker:Richard M. Karp (University of Washington)

Title:Algorithms for Graph Partitioning on the Planted Partition Model

Abstract:The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph k-partition problem is to partition the nodes of an undirected graph into k equal-sized groups so as to minimize the total number of edges that cross between groups.

We present a simple, linear-time algorithm for the graph k-partition problem and analyze it on a random ``planted'' k-partition model. In this model, the n nodes of a graph are partitioned into k groups, each of size n/k; two nodes in the same group are connected by an edge with some probability $p$, and two nodes in different groups are connected by an edge with some probability $r < p$. We show that if $p-r is sufficiently large then the algorithm finds the optimal partition with high probability. Joint work with Anne Condon of the University of Wisconsin.

Time:Monday, April 5, 1999 at 2:30 p.m.

Location:LOW 216

Speaker:Zhen-Qing Chen (University of Washington)

Title:Green function and Poisson kernel estimates for symmetric stable processes

Abstract:In this talk, we will present some new two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric $\alpha$-stable process in bounded $C^{1,1}$ open sets. The new estimates give explicit information on how the comparing constants depend on parameter $\alpha$ and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing $\alpha$ to 2.

Time:Wednesday, March 3, 1999 at 2:30 p.m.

Location:CMU B006

Speaker:Ron Pyke (University of Washington)

Title:ON THE SOLIDARITY OF RECURRENT MARKOV RENEWAL PROCESSES

Abstract:It is known that for a wide class of irreducible recurrent Markov Chains and Markov Renewal processes, the finiteness of the $p$-th moment of recurrence times is finite for one state if and only if it is finite for all states. This result also holds for a wide class of functionals of the processes between successive occurrence times. Such results are known as solidarity theorems. In this paper, a solidarity theorem is proved concerning the property of ``being in the domain of attraction of a stable law with specified parameters.'' A basic identity linking additive random functions determined by two distinct states is emphasized, and used to derive solidarity results for moments and for the WLLN.

Time:Wednesday, February 24, 1999 at 2:30 p.m.

Location:CMU B006

Speaker:Ron Pyke (University of Washington)

Title:REASSEMBLING OF SHATTERED BROWNIAN SHEET

Abstract:If $D_n:n\ge 1$ is a given countable collection of Borel subsets of the unit square $[0, 1]^2=I^2$ and $Z$ is a standard Brownian sheet over $I^2$, is it possible to reconstruct $Z$ from the knowledge of all of the patches $Z-c_n$ over transformed domains $\tau_n(D_n)$ for unknown constants $c_n$ and unknown rotation-translation transformations $\tau_n$? We show that the answer to this question is yes under fairly natural restrictions on the sets $D_n$. The main property of Brownian sheet that leads to this possibility is that the local behaviour of $Z$ around a point $\bt$ actually determines $\bt$. In this sense, a Brownian sheet carries with it its own location coordinates. An open problem exists when the fragments are rotated in their range space.

Mathematics Department Colloquium

Time:Tuesday, February 9, 1999 at 4:00 p.m.

Location:CMU 120

Speaker:Persi Diaconis (Stanford University)

Title:MATHEMATICS OF SOLITAIRE

Abstract:One of the embarrassment of applied probability is that we can not analyze the original game of solitaire. I will show that a simplified solitaire leads to fascinating mathematics and complete analysis. The mathematics involves combinatorics, random matrices, and Riemann-Hilbert theorem. This is a joint work with David Aldous.

This coloquium talk is accessible to general audience including undergraduate students.

Time:Tuesday, February 9, 1999 at 2:30 p.m.

Location:Padelford C-401

Speaker:Amir Dembo (Stanford University)

Title:THICK, THIN AND STICKY POINTS OF BROWNIAN MOTION

Abstract:We find the correct nondegenerate ``multifractal spectrum'' for Brownian occupation measure in ${\bf R}^d$, $d \geq 2$. It involves {\it thick points} $x$ for which the occupation measures of small balls (radius $r$) centered at $x$ are exceptionally large {\it for some} $r_n \downarrow 0$; {\it sticky points} having such a property {\it for all} $r \downarrow 0$ and {\it thin points} with exceptionally small occupation measures for some $r_n \downarrow 0$. As a by product we resolve problems posed by Taylor (1974) (for $d \geq 3$) and Perkins and Taylor (1987) (for $d=2$), as well as Erd\"{o}s and Taylor (1960) long standing open problem about favorite points for the simple random walk in ${\bf Z}^2$.

Our results are related to the LILs of Ciesielski \& Taylor (1962) and Ray (1963) for the Brownian occupation measure of small balls, in the same way that L\'{e}vy's uniform modulus of continuity, and the formula of Orey and Taylor (1974) for the dimension of {\it fast points}, are related to the usual LIL.

In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of `limsup type'.

This talk is based on a joint work with Y. Peres, J. Rosen and O. Zeitouni.

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