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Probability seminar
- 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.
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