# Probability Seminar

# Fall 2016

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

**
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**

## Thursday, January 28, Leonid Petrov, University of Virginia

Title: **The quantum integrable particle system on the line**

## Thursday, February 25, Ramon van Handel, ORFE and PACM, Princeton

Title: **The norm of structured random matrices**

Abstract: Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure and applied mathematics. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to easily address otherwise nontrivial phenomena such as the phase transition of the spectral edge of random band matrices. I will also discuss some conjectures whose resolution would complete our understanding of the underlying probabilistic mechanisms.

## Thursday, March 3, Chris Janjigian, UW-Madison

Title: **Large deviations for certain inhomogeneous corner growth models**

Abstract: The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.

This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)

## Thursday, March 10, Jun Yin, UW-Madison

Title: **Delocalization and Universality of band matrices.**

Abstract: in this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are widely believed to have the same asymptotic behaviors as those of Wigner matrices. We proved that this conjecture is true as long as the bandwidth is wide enough.

## Thursday, March 17, Sebastien Roch, UW-Madison

Title: **Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer**

Abstract Reconstructing the tree of life from molecular sequences is a fundamental problem in computational biology. Modern data sets often contain large numbers of genes. That can complicate the reconstruction because different genes often undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In this talk I will sketch recent results showing that, under a natural stochastic model of LGT, the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT.

## Thursday, March 24, No Seminar, Spring Break

## Thursday, March 31, Bill Sandholm, Economics, UW-Madison

Title: **A Sample Path Large Deviation Principle for a Class of Population Processes**

Abstract: We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cramer transform L(·,·), the running cost appearing in the large deviation principle rate function.

## Thursday, April 7, No Seminar

## Thursday, April 14, Jessica Lin, UW-Madison, Joint with PDE Geometric Analysis seminar

Title: **Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form**

Abstract: I will present optimal quantitative error estimates in the stochastic homogenization for uniformly elliptic equations in nondivergence form. From the point of view of probability theory, stochastic homogenization is equivalent to identifying a quenched invariance principle for random walks in a balanced random environment. Under strong independence assumptions on the environment, the main argument relies on establishing an exponential version of the Efron-Stein inequality. As an artifact of the optimal error estimates, we obtain a regularity theory down to microscopic scale, which implies estimates on the local integrability of the invariant measure associated to the process. This talk is based on joint work with Scott Armstrong.

## Thursday, April 21, Paul Bourgade, Courant Institute, NYU

Title: **Freezing and extremes of random unitary matrices**

Abstract: A conjecture of Fyodorov, Hiary & Keating states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the log-correlated Gaussian fields. We will outline the proof of the conjecture for the leading order of the maximum, and a freezing of the free energy related to the matrix model. This talk is based on a joint work with Louis-Pierre Arguin and David Belius.

## Thursday, April 28, Nancy Garcia, Statistics, IMECC, UNICAMP, Brazil

Title: **Rumor processes on [math]\mathbb{N}[/math] and discrete renewal processe**

Abstract: We study two rumor processes on the positive integers, the dynamics of which are related to an SI epidemic model with long range transmission. Start with one spreader at site [math]0[/math] and ignorants situated at some other sites of [math]\mathbb{N}[/math]. The spreaders transmit the information within a random distance on their right. Depending on the initial distribution of the ignorants, we obtain probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our approach is to relate this model to the house-of-cards.

## Thursday, May 5, Diane Holcomb, University of Arizona

Title: **Local limits of Dyson's Brownian Motion at multiple times**

Abstract: Dyson's Brownian Motion may be thought of as a generalization of Brownian Motion to the matrix setting. We can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).

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