# Probability Seminar

## Spring 2012

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit this page to sign up for the email list.

## Thursday, January 26, Timo Seppäläinen, University of Wisconsin - Madison

Title: The exactly solvable log-gamma polymer

Abstract: Among 1+1 dimensional directed lattice polymers, log-gamma distributed weights are a special case that is amenable to various useful exact calculations (an *exactly solvable* case). This talk discusses various aspects of the log-gamma model, in particular an approach to analyzing the model through a geometric or "tropical" version of the Robinson-Schensted-Knuth correspondence.

## Thursday, February 9, Arnab Sen, Cambridge

Title: Random Toeplitz matrices

Abstract: Random Toeplitz matrices belong to the exciting area that lies at the intersection of the usual Wigner random matrices and random Schrodinger operators. In this talk I will describe two recent results on random Toeplitz matrices. First, the maximum eigenvalue, suitably normalized, converges to the 2-4 operator norm of the well-known Sine kernel. Second, the limiting eigenvalue distribution is absolutely continuous, which partially settles a conjecture made by Bryc, Dembo and Jiang (2006). I will also present several open questions and conjectures.

This is a joint work with Balint Virag (Toronto).

## Thursday, February 16, Benedek Valko, University of Wisconsin - Madison

Title: Point processes and carousels

Abstract: For several classical matrix models the joint density of the eigenvalues can be written as an expression involving a Vandermonde determinant raised to the power of 1, 2 or 4. Most of these examples have beta-generalizations where this exponent is replaced by a parameter beta>0. In recent years the point process limits of various beta ensembles have been derived. The limiting processes are usually described as the spectrum of certain stochastic operators or with the help of a coupled system of SDEs. In the bulk beta Hermite case (which is the generalization of GUE) there is a nice geometric construction of the point process involving a Brownian motion in the hyperbolic plane, this is the Brownian carousel. Surprisingly, there are a number of other limit processes that have carousel like representation. We will discuss a couple of examples and some applications of these new representations.

Joint with Balint Virag.

## Thursday, February 23, Tom Kurtz, University of Wisconsin - Madison

Title: Particle representations for SPDEs and strict positivity of solutions

Abstract: Stochastic partial differential equations arise naturally as limits of finite systems of interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. The support properties of the measure-valued solution can be studied using Girsanov change of measure techniques. The ideas will be illustrated by a model of asset prices set by an infinite system of competing traders. These latter results are joint work with Dan Crisan and Yoonjung Lee.

## Wednesday, February 29, 2:30pm, Scott Armstrong, University of Wisconsin - Madison

VV B309

Title: PDE methods for diffusions in random environments

Abstract: I will summarize some recent work with Souganidis on the stochastic homogenization of (viscous) Hamilton-Jacobi equations. The homogenization of (special cases of) these equations can be shown to be equivalent to some well-known results of Sznitman in the 90s on the quenched large deviations of Brownian motion in the presence of Poissonian obstacles. I will explain the PDE point of view and speculate on some further connections that can be made with probability.

## Wednesday, March 7, 2:30pm, Paul Bourgade, Harvard

VV B309

Title: Universality for beta ensembles.

Abstract: Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erdos and H.-T. Yau, which yields local universality for the log-gases at arbitrary temperature, for analytic external potential. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

## Thursday, March 8, William Stanton, UC Boulder

Title: An Improved Right-Tail Upper Bound for a KPZ "Crossover Distribution"

Abstract: In the last decade, there has been an explosion of research on KPZ universality: the notion that the Kardar-Parisi-Zhang stochastic PDE from statistical physics describes the fluctuations of a large class of models. In 2010, Amir, Corwin, and Quastel proved that the KPZ equation arises as a scaling limit of the Weakly Asymmetric Simple Exclusion Process (WASEP). In this sense, the KPZ equation interpolates between the KPZ universality class and the Edwards-Wilkinson class. Thus, the distributions of height functions of the KPZ equation are called the "crossover distributions." In this talk, I will introduce the notion of KPZ universality and the crossover distributions and present a new result giving an improved upper bound for the right-tail of a particular crossover distribution.

## Friday, April 13, 2:30pm, Gregory Shinault, UC Davis

VV B305

Title: Inhomogeneous Tilings of the Aztec Diamond

Random domino tilings of the Aztec diamond have been a subject of much interest in the past 20 years. The main result of the subject, the Arctic Circle theorem, is a gem of modern mathematics which gives the limiting shape of the tiling. When we examine fluctuations around the limiting shape, we do not see a Gaussian distribution as one might expect in classical probability. Instead we see the Tracy-Widom GUE distribution of random matrix theory. These theorems were originally proven for a random tiling chosen by a uniform distribution. In this talk we examine the effects of choosing the tiling via a distribution in an inhomogeneous environment (and we'll explain what we mean by this!).

## Thursday, April 19, Nancy Garcia, Universidade Estadual de Campinas

Title: Perfect simulation for chains and processes with infinite range

Abstract: In this talk we discuss how to perform Kalikow-type decompositions for discontinuous chains of infinite memory and for interacting particle systems with interactions of infinite range. Then, we will show how this decomposition can be used to generate samples from these systems.

## Thursday, April 26, Jim Kuelbs, University of Wisconsin - Madison

A CLT for Empirical Processes and Empirical Quantile Processes Involving Time Dependent Data

We establish empirical quantile process CLT's based on [math]n[/math] independent copies of a stochastic process [math]\{X_t: t \in E\}[/math] that are uniform in [math]t \in E[/math] and quantile levels [math]\alpha \in I[/math], where [math]I[/math] is a closed sub-interval of [math](0,1)[/math]. Typically [math]E=[0,T][/math], or a finite product of such intervals. Also included are CLT's for the empirical process based on [math]\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}[/math] that are uniform in [math]t \in E, y \in R[/math]. The process [math]\{X_t: t \in E\}[/math] may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.

## Monday, April 30, 2:30pm, Lukas Szpruch, Oxford

VV B337

Title: Antithetic multilevel Monte Carlo method.

We introduce a new multilevel Monte Carlo
(MLMC) estimator for multidimensional SDEs driven by Brownian motion.
Giles has previously shown that if we combine a numerical approximation
with strong order of convergence $O(\D t)$ with MLMC we can reduce
the computational complexity to estimate expected values of
functionals of SDE solutions with a root-mean-square error of $\eps$
from $O(\eps^{-3})$ to $O(\eps^{-2})$. However, in general, to obtain
a rate of strong convergence higher than $O(\D t^{1/2})$ requires
simulation, or approximation, of \Levy areas. Through
the construction of a suitable antithetic multilevel correction estimator,
we are able to avoid the simulation of \Levy areas and still achieve an
$O(\D t2)$ variance for smooth payoffs, and almost an $O(\D t^{3/2})$ variance for
piecewise smooth payoffs,
even though there is only $O(\D t^{1/2})$ strong convergence. This
results in an $O(\eps^{-2})$ complexity for estimating the
value of European and Asian put and call options.
We also comment on the extension of the antithetic
approach to pricing Asian and barrier options.

## Wednesday, May 2, 2:30pm, Wenbo Li, University of Delaware

VV B337

Title: Probabilities of all real zeros for random polynomials

Abstract: There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions. The talk is accessible to undergraduate and graduate students in any areas of mathematics.

## Thursday, May 3, Samuel Isaacson, Boston University

Title: Relationships between several particle-based stochastic reaction-diffusion models.

Abstract: Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.