Probability seminar
 Title: COLLISIONS OF RANDOM WALKS ON SOME GRAPHS Speaker: Xinxing Chen (Shanghai Jiaotong University) Time: 2:30 p.m., Monday, March 11, 2013 Room: MGH 242 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.     Title: JOINT CONVERGENCE ALONG DIFFERENT SUBSEQUENCES OF THE SIGNED CUBIC VARIATION OF FRACTIONAL BROWNIAN MOTION Speaker: Jason Swanson (University of Central Florida) Time: 2:30 p.m., Wednesday, March 6, 2013 Room: LOW 220 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, $\Delta t$, 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 $\Delta t$.     Title: DIFFUSIVE LIMITS OF SOME INTERACTING PARTICLE SYSTEMS OF EXCLUSION TYPE Speaker: Mykhaylo Shkolnikov (University of California, Berkeley) Time: 2:30 p.m., Monday, March 4, 2013 Room: MGH 242 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.     Title: BOUNDARY HARNACK PRINCIPLE AND MARTIN BOUNDARY AT INFINITY FOR SUBORDINATE BROWNIAN MOTIONS Speaker: Panki Kim (Seoul National University) Time: 2:30 p.m., Monday, February 25, 2013 Room: MGH 242 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     Title: THE BROWNIAN CONGA LINE Speaker: Sayan Banerjee (University of Washington) Time: 2:30 p.m., Wednesday, February 20, 2013 Room: MGH 242 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.     Title: STRONG APPROXIMATIONS TO A QUANTILE PROCESS BASED ON $n$ INDEPENDENT FRACTIONAL BROWNIAN MOTIONS Speaker: David M. Mason (University of Delaware) Time: 2:30 p.m., Monday, February 11, 2013 Room: MGH 242 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.     Title: GIANT COMPONENTS IN RANDOM GRAPHS WITH RANDOM VERTEX SETS Speaker: Mary Radcliffe (University of Washington) Time: 2:30 p.m., Monday, February 4, 2013 Room: MGH 242 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.     Title: DETERMINANT OF LAPLACIANS AND RANDOM MULTICURVES ON SURFACES Speaker: Adrien Kassel (Ecole Normale Superieure) Time: 2:30 p.m., Monday, January 28, 2013 Room: MGH 242 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 $\Omega$ 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 $\Omega$. Joint work with Rick Kenyon.     Title: RANDOM 312 AVOIDING PERMUTATIONS Speaker: Lerna Pehlivan (York University, Toronto) Time: 2:30 p.m., Monday, January 14, 2013 Room: MGH 242 Abstract: A permutation of $\{1,2,\dots ,N\}$ 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.     Title: THE AZTEC DIAMOND AND KPZ UNIVERSALITY Speaker: Mauricio Duarte (Universidad de Chile) Time: 2:30 p.m., Monday, January 7, 2013 Room: MGH 242 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.     Title: A GEOMETRIC PERSPECTIVE ON FIRST-PASSAGE COMPETITION Speaker: Nathaniel Blair-Stahn (University of Washington) Time: 2:30 p.m., Monday, May 21, 2012 Room: SMI 211 Abstract: First-passage competition is a stochastic process modeling two species competing for space in the integer lattice $\mathbf{Z}^d$, 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 Häggström and Pemantle in 1998 as a generalization of first-passage percolation, which models a single species spreading throughout the graph $\mathbf{Z}^d$. 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 $\mathbf{Z}^d$, 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 $\mathbf{R}^d$, 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 Häggström and Pemantle regarding survival of the two species when each starts at a single point.     Title: CONNECTIONS BETWEEN THE ABELIAN SANDPILE MODEL AND THE DIMER MODEL Speaker: Laura Florescu (Los Alamos National Lab) Time: 2:30 p.m., Monday, May 14, 2012 Room: SMI 211 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.     Title: SPINNING BROWNIAN MOTION Speaker: Mauricio Duarte (University of Washington) Time: 2:30 p.m., Monday, May 7, 2012 Room: SMI 211 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.     Title: HYDRODYNAMIC LIMIT OF A BOUNDARY-DRIVEN ELASTIC EXCLUSION PROCESS AND A STEFAN PROBLEM Speaker: Joel Barnes (University of Washington) Time: 2:30 p.m., Monday, April 30, 2012 Room: SMI 211 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.     Title: EVOLUTION FROM SEEDS IN ONE-DIMENSIONAL CELLULAR AUTOMATA Speaker: Janko Gravner (University of California, Davis) Time: 2:30 p.m., Monday, April 23, 2012 Room: SMI 211 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.)     Title: EFFECTIVE DYNAMICS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Speaker: Jinqiao Duan (Institute for Pure and Applied Mathematics (IPAM) and Illinois Institute of Technology) Time: 2:30 p.m., Monday, April 16, 2012 Room: SMI 211 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.     Title: EFFECT OF SELECTION ON THE GENEALOGY OF POPULATIONS Speaker: Nathanael Berestycki (Cambridge University) Time: 2:30 p.m., Monday, April 9, 2012 Room: SMI 211 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).     Title: THE SCALING LIMIT OF THE MINIMAL SPANNING TREE ON THE COMPLETE GRAPH Speaker: Gregory Miermont (Universite Paris-Sud, Orsay) Time: 2:30 p.m., Monday, April 2, 2012 Room: SMI 211 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.     Title: GROWING RANDOM REGULAR GRAPHS AND THE GAUSSIAN FREE FIELD Speaker: Toby Johnson (University of Washington) Time: 2:30 p.m., Monday, March 26, 2012 Room: SMI 211 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.
Last modified: March 7, 2013, 09:37