# Probability Seminar

# Spring 2015

**Thursdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.

**
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**

## Thursday, January 15, Miklos Racz, UC-Berkeley Stats

Title: Testing for high-dimensional geometry in random graphs

Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.

## Thursday, January 22, No Seminar

## Thursday, January 29, Arnab Sen, University of Minnesota

Title: **Double Roots of Random Littlewood Polynomials**

Abstract: We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions.

This is joint work with Ron Peled and Ofer Zeitouni.

## Thursday, February 5, No seminar this week

## Thursday, February 12, No Seminar this week

## Thursday, February 19, Xiaoqin Guo, Purdue

Title: Quenched invariance principle for random walks in time-dependent random environment

Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in [math]Z^d[/math]. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.

## Thursday, February 26, Dan Crisan, Imperial College London

Title: **Smoothness properties of randomly perturbed semigroups with application to nonlinear filtering**

Abstract: In this talk I will discuss sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. The estimates we derive have sharp small time asymptotics

This is joint work with Terry Lyons (Oxford) and Christian Literrer (Ecole Polytechnique) and is based on the paper

D Crisan, C Litterer, T Lyons, Kusuoka–Stroock gradient bounds for the solution of the filtering equation, Journal of Functional Analysis, 2105

## Wednesday, March 4, Sam Stechmann, UW-Madison, 2:25pm Van Vleck B113

Please note the unusual time and room.

Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena

Abstract:
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.

## Thursday, March 12, Ohad Feldheim, IMA

Title: **The 3-states AF-Potts model in high dimension**

Abstract: Take a bounded odd domain of the bipartite graph [math]\mathbb{Z}^d[/math]. Color the boundary of the set by [math]0[/math], then color the rest of the domain at random with the colors [math]\{0,\dots,q-1\}[/math], penalizing every configuration with proportion to the number of improper edges at a given rate [math]\beta\gt 0[/math] (the "inverse temperature"). Q: "What is the structure of such a coloring?"

This model is called the [math]q[/math]-states Potts antiferromagnet(AF), a classical spin glass model in statistical mechanics. The [math]2[/math]-states case is the famous Ising model which is relatively well understood. The [math]3[/math]-states case in high dimension has been studies for [math]\beta=\infty[/math], when the model reduces to a uniformly chosen proper three coloring of the domain. Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the structure of the model showing long-range correlations and phase coexistence. In this work, we generalize this result to positive temperature, showing that for large enough [math]\beta[/math] (low enough temperature) the rigid structure persists. This is the first rigorous result for [math]\beta\lt \infty[/math].

In the talk, assuming no acquaintance with the model, we shall give the physical background, introduce all the relevant definitions and shed some light on how such results are proved using only combinatorial methods. Joint work with Yinon Spinka.

## Thursday, March 19, Mark Huber, Claremont McKenna Math

Title: Understanding relative error in Monte Carlo simulations

Abstract: The problem of estimating the probability [math]p[/math] of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem. In this talk, I'll consider a new twist: given an estimate [math]\hat p[/math], suppose we want to understand the behavior of the relative error [math](\hat p - p)/p[/math]. In classic estimators, the values that the relative error can take on depends on the value of [math]p[/math]. I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of [math]p[/math]. Moreover, this new estimate is very fast: it takes a number of coin flips that is very close to the theoretical minimum. Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.

## Thursday, March 26, Ji Oon Lee, KAIST

Title: Tracy-Widom Distribution for Sample Covariance Matrices with General Population

Abstract: Consider the sample covariance matrix [math](\Sigma^{1/2} X)(\Sigma^{1/2} X)^*[/math], where the sample [math]X[/math] is an [math]M \times N[/math] random matrix whose entries are real independent random variables with variance [math]1/N[/math] and [math]\Sigma[/math] is an [math]M \times M[/math] positive-definite deterministic diagonal matrix. We show that the fluctuation of its rescaled largest eigenvalue is given by the type-1 Tracy-Widom distribution. This is a joint work with Kevin Schnelli.

## Thursday, April 2, No Seminar, Spring Break

## Thursday, April 9, Elnur Emrah, UW-Madison

Title: The shape functions of certain exactly solvable inhomogeneous planar corner growth models

Abstract: I will talk about two kinds of inhomogeneous corner growth models with independent waiting times {W(i, j): i, j positive integers}: (1) W(i, j) is distributed exponentially with parameter [math]a_i+b_j[/math] for each i, j.(2) W(i, j) is distributed geometrically with fail parameter [math]a_ib_j[/math] for each i, j. These generalize exactly-solvable i.i.d. models with exponential or geometric waiting times. The parameters (a_n) and (b_n) are random with a joint distribution that is stationary with respect to the nonnegative shifts and ergodic (separately) with respect to the positive shifts of the indices. Then the shape functions of models (1) and (2) satisfy variational formulas in terms of the marginal distributions of (a_n) and (b_n). For certain choices of these marginal distributions, we still get closed-form expressions for the shape function as in the i.i.d. models.

## Thursday, April 16, Scott Hottovy, UW-Madison

Title: **An SDE approximation for stochastic differential delay equations with colored state-dependent noise**

Abstract: In this talk I will introduce a stochastic differential delay equation with state-dependent colored noise which arises from a noisy circuit experiment. In the experimental paper, a small delay and correlation time limit was performed by using a Taylor expansion of the delay. However, a time substitution was first performed to obtain a good match with experimental results. I will discuss how this limit can be proved without the use of a Taylor expansion by using a theory of convergence of stochastic processes developed by Kurtz and Protter. To obtain a necessary bound, the theory of sums of weakly dependent random variables is used. This analysis leads to the explanation of why the time substitution was needed in the previous case.

## Thursday, April 23, Hoi Nguyen, Ohio State University

Title: On eigenvalue repulsion of random matrices

Abstract:

I will address certain repulsion behavior of roots of random polynomials and of eigenvalues of Wigner matrices, and their applications. Among other things, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for quite general matrix ensembles.

## Thursday, May 7, Jessica Lin, UW-Madison

Title: **Random Walks in Random Environments and Stochastic Homogenization**

In this talk, I will draw connections between random walks in random environments (RWRE) and stochastic homogenization of partial differential equations (PDE). I will introduce various models of RWRE and derive the corresponding PDEs to show that the two subjects are intimately related. I will then give a brief overview of the tools and techniques used in both approaches (reviewing some classical results), and discuss some recent problems in RWRE which are related to my research in stochastic homogenization.

## Thursday, May 14, Chris Janjigian, UW-Madison

Title: **Large deviations of the free energy in the O’Connell-Yor polymer**

Abstract: The first model of a directed polymer in a random environment was introduced in the statistical physics literature in the mid 1980s. This family of models has attracted substantial interest in the mathematical community in recent years, due in part to the conjecture that they lie in the KPZ universality class. At the moment, this conjecture can only be verified rigorously for a handful of exactly solvable models. In order to further explore the behavior of these models, it is natural to question whether the solvable models have any common features aside from the Tracy-Widom fluctuations and scaling exponents that characterize the KPZ class.

This talk considers the behavior of one of the solvable polymer models when it is far away from the behavior one would expect based on the KPZ conjecture. We consider the model of a 1+1 dimensional directed polymer model due to O’Connell and Yor, which is a Brownian analogue of the classical lattice polymer models. This model satisfies a strong analogue of Burke’s theorem from queueing theory, which makes some objects of interest computable. This talk will discuss how, using the Burke property, one can compute the positive moment Lyapunov exponents of the parabolic Anderson model associated to the polymer and how this leads to a computation of the large deviation rate function with normalization n for the free energy of the polymer.