# Difference between revisions of "Probability Seminar"

## Friday, September 3, 4PM B239 Timo Seppäläinen (UW Madison) (Math Colloquium)

Title: Scaling exponents for a 1+1 dimensional directed polymer
Abstract: Directed polymer in a random environment is a model from statistical physics that has been around for 25 years. It is a type of random walk that evolves in a random potential. This means that the walk lives in a random landscape, some parts of which are favorable and other parts unfavorable to the walk. The objective is to understand the behavior of the walk on large space and time scales.
I will begin the talk with simple random walk straight from undergraduate probability and explain what diffusive behavior of random walk means and how Brownian motion figures into the picture. The recent result of the talk concerns a particular 1+1 dimensional polymer model: the order of magnitude of the fluctuations of the polymer path is described by the exponent 2/3, in contrast with the exponent 1/2 of diffusive paths. Finding a rigorous proof of this exponent has been an open problem since the introduction of the model.

## Thursday, September 16, Gregorio Moreno Flores (UW - Madison)

Title: Asymmetric directed polymers in random environments.
Abstract: It is well known that the asymmetric last passage percolation problem can be approximated by a Brownian percolation model, which is itself related to the GUE random matrices. This allows to transfer many results about random matrices to the setting of asymmetric last passage percolation.
In this talk, we will introduce two different schemes to treat asymmetric directed polymers in random environments.

## Thursday, September 30, Brian Rider (University of Colorado at Boulder)

Title: Solvable two-charge models
Abstract: I'll describe recent progress on ensembles of random matrix type which can be viewed as having particles of two distinct "charges", subject to coulombic interaction. The natural (and classic) example is Ginibre's non-symmetric Gaussian matrix in which the particles (eigenvalues) live in the complex plane. Taking this as a starting point and forcing the particles down to the line produces a family of ensembles which interpolate (though not in the way we might want) between the well studied Gaussian Orthogonal and Symplectic Ensembles.
Joint work with Christopher Sinclair and Yuan Xu (Univ. Oregon).

## Thursday, October 7, Benedek Valko (UW - Madison)

Title: Scaling limits of tridiagonal matrices
Abstract: I will describe the point process limits of the spectrum for a certain class of tridiagonal matrices. The limiting point process can be defined through a coupled system of stochastic differential equations. I will discuss various applications of this description, e.g. eigenvalue repulsion, probability of large gaps and central limit theorem for the number of points in an interval.

Joint work with E. Kritchevski and B. Virag (Toronto).

## Thursday, October 21, Jim Kuelbs (UW - Madison)

Title: An Empirical Process CLT for Time Dependent Data
Abstract: For stochastic processes $\{X_t: t \in E\}$, we establish sufficient conditions for the empirical process based on $\{ I_{X_t \le y} - P(X_t \le y): t \in E, y \in \mathbb{R}\}$ to satisfy the CLT uniformly in $t \in E, y \in \mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E= [0,1]$.
Joint work with Tom Kurtz and Joel Zinn.

Title: TBA

## Wednesday, November 17, 2:30pm, Philip Matchett Wood (Stanford)

NOTE THE UNUSUAL TIME!

Title: Random tridiagonal doubly stochastic matrices
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 ‘typical’ 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.

Title: TBA