# Probability Seminar

# Fall 2015

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

**
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**

## Thursday, September 17, Nicholas A. Cook, UCLA, 2:25pm Van Vleck B325

** Please note the unusual location, Van Vleck Hall B325 **

Title: **Random regular digraphs: singularity and spectrum**

We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the adjacency matrix without iid weights, we prove that it is invertible with high probability. Along the way we make use of Stein's method of exchangeable pairs to establish some graph discrepancy properties.

## Thursday, September 24, No seminar

## Thursday, October 1 Sebastien Roch, UW-Madison

Title: **Mathematics of the Tree of Life--From Genomes to Phylogenetic Trees and Beyond**

Abstract: The reconstruction of the Tree of Life is an old problem in evolutionary biology which has benefited from various branches of mathematics, including probability, combinatorics, algebra, and geometry. Modern DNA sequencing technologies are producing a deluge of new data on a vast array of organisms--transforming how we view the Tree of Life and how it is reconstructed. I will survey recent progress on some mathematical and computational questions that arise in this context. No biology background will be assumed. (This is a practice run for a plenary talk at an AMS meeting.)

## Thursday, October 8, No Seminar due to the Midwest Probability Colloquium

Midwest Probability Colloquium

## Thursday, October 15, Louis Fan, UW-Madison

Title: **Reflected diffusions with partial annihilations on a membrane (Part two)**

Abstract: Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce an interacting particle system used to model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales, we show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is the second part of a previous talk given in the Applied and Computation math seminar. Our proofs are based on a correlation function technique (studying the BBGKY hierarchy) and its generalization. This is joint work with Zhen-Qing Chen.

## Thursday, October 22, Tom Kurtz, UW-Madison

Title: **Strong and weak solutions for general stochastic models**

Abstract: Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls to stochastic “outputs.” A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations will be given in this context. A notion of “compatibility” between inputs and outputs is critical in relating the general result to its classical forebears. Time-change equations for diffusion processes provide an interesting example. Such equations arise naturally as limits of analogous equations for Markov chains. For one-dimensional diffusions they also are essentially given in the now-famous notebook of Doeblin. Although requiring nothing more than standard Brownian motions and the Riemann integral to define, the question of strong uniqueness remains unresolved. To prove weak uniqueness, the notion of compatible solution is employed and the martingale properties of compatible solutions used to reduce the uniqueness question to the corresponding question for a martingale problem or an Ito equation.

## Thursday, October 29, Ecaterina Sava-Huss, Cornell

Title: **Interpolating between rotor walk and random walk**

Abstract: After a short introduction on deterministic random walks (called also rotor-router walks) and some related cluster growth models, I will introduce a family of stochastic processes on the integers, depending on a parameter p. These processes interpolate between the deterministic rotor walk (for p=0) and the simple random walk (for p=1/2), and they are not Markovian. For such processes, I will prove that the scaling limit is a one-sided perturbed Brownian motion, which is a linear combination of a Brownian motion and its running maximum. This is based on joint work with Wilfried Huss and Lionel Levine.

## Thursday, November 5, No Seminar this week

## Thursday, November 12, SEMINAR CANCELLED

Thunderstorms in Chicago an 11/11 cancelled the speaker's flights; we will try to re-schedule.

## Thursday, November 19, David Herzog Iowa State

Title: **Stabilization by noise and the existence of optimal Lyapunov functions**

Abstract: We discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.

## Thursday, November 26, No Seminar, Thanksgiving Break

## Thursday, December 3, Janna Lierl, UIUC

Title: **Parabolic Harnack inequality on fractal Dirichlet spaces**

Abstract: I will present some recent results on extending the parabolic Moser iteration method to the setting of (fractal-type) metric measure Dirichlet spaces. Under appropriate geometric conditions, we obtain that local weak solutions to heat equation are locally bounded, H\"older continuous, and satisfy a strong parabolic Harnack inequality. If time permits, I will also discuss the case of time-dependent Dirichlet forms, or non-symmetric perturbations of the Dirichlet form. Applications of the parabolic Harnack inequality include sharp upper and lower bounds for the associated heat kernel.

## Thursday, December 10, Elizabeth Meckes, Case Western Reserve University

Title: **Patterns in Eigenvalues: Random matrices from the compact classical groups.**

Abstract: There are many striking features of the eigenvalues of random orthogonal and unitary matrices. In this talk, I'll describe Haar measure on those groups and the resulting distributions of the eigenvalues. I will give a survey of now-classical asymptotic results, and then describe a result of E. Rains and a recent result of mine (joint with M. Meckes), which demonstrate some intriguing self-similarities of the eigenvalue processes. Prerequisites will be kept to a minimum.

## Spring Semester

A selection of the spring semester schedule starts below