# 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.

## Thursday, October 28, Tom Alberts (University of Toronto)

Title: Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer
Abstract: The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature $\beta$ the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive $\beta$ are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length $n$ of the polymer. The precise scaling is $\beta n^{-1/4}$. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension.
Joint work with Kostya Khanin and Jeremy Quastel.

## Wednesday, November 17, 2:30pm, B239 VV, Philip Matchett Wood (Stanford)

NOTE THE UNUSUAL TIME AND PLACE!

Title: Singularity of Discrete Random Matrices
Consider an n by n square matrix where n is large. For each entry, flip a fair coin, making the entry +1 if the coin comes up heads, and -1 if the coin comes up tails. What is the probability that the matrix has determinant equal to zero? This talk will discuss work that builds on a breakthrough by Terence Tao and Van Vu in 2007 who used ideas in combinatorics, probability theory, and additive combinatorics to prove upper bounds on the probability of singularity. New results allow us to work with more general random matrices and prove better upper bounds. Joint work with Jean Bourgain and Van Vu.

## Thursday, November 18, Joseph S. Miller (UW - Madison)

Title: An introduction to algorithmic randomness
Abstract: Various attempts have been made to give meaning to the idea that an individual binary sequence is random, starting with Von Mises in 1919. He gave the first published axiomatization of probability theory, basing it on a distinguished family of random sequences. The modern approach to defining randomness for individual sequences is rooted Kolmogorov's definition of the complexity of a finite binary string. Kolmogorov complexity is closely related to the most robust notion of (sufficient) randomness for infinite binary sequences, given by Martin-Löf. I will introduce these notions and talk about how they interact with computability theory (my field of study), analysis, and our intuitions about randomness.

## Thursday, December 2, Hao Lin (UW - Madison)

Title: Properties of the limit shape for some last passage growth models in random environments
We study directed last passage percolation on the first quadrant of the planar square lattice whose weights have general distributions, or equivalently, ./G/1 queues in series. The service time distributions of the servers vary randomly which constitutes a random environment for the model. Equivalently, each row of the last passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random means attached to the rows.