# Probability Seminar

## Fall 2012

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit this page to sign up for the email list.

## Thursday, September 13, Sebastien Roch, UW-Madison

Title: Markov models on trees: Variants of the reconstruction problem

Abstract: I will consider the so-called `reconstruction problem': how accurately can one guess the state at the root of a Markov chain on a finite tree, given the states at the leaves? I will introduce variants of this problem that arise naturally in connection with applications in molecular evolution, and discuss recent results and open problems. Based on joint works with Andoni, Daskalakis, Hassidim, Mossel and Sly.

## Thursday, September 20, Jun Yin, UW-Madison

Title: Some new results on random matrices.

Abstract: In this talk, we will introduce some new results on random matrices, especially the necessary and sufficient conditions for universality at the edge and a new result on the circular law.

## Friday, October 5, Nicos Georgiou, University of Utah

Title: Busemann functions and variational formula for last passage percolation.

Abstract: Directed last passage percolation on the two dimensional lattice is exactly solvable when the weight distribution is i.i.d. exponential or geometric. The reason for that is the Burke property associated to a model with "boundaries".

We investigate the solvable model further in order to generalize the idea of boundaries into the general setting, and we compute a variational formula for passage times for more general weights. The variatonal formula is given in terms of Busemann functions and all restrictive assumptions on the environment are to guarantee their existence.

Joint work with T. Seppalainen, F. Rassoul-Agha and A. Yilmaz.

## Thursday, October 11, No seminar

because of the MIDWEST PROBABILITY COLLOQUIUM

## Thursday, October 18, Jason Swanson, University of Central Florida

Title: Correlations within the signed cubic variation of fractional Brownian motion

Abstract: The signed cubic variation of the fractional Brownian motion, [math]B[/math], with Hurst parameter [math]H=1/6[/math], is a concept built upon the fact that the sequence, [math]\{W_n\}[/math], of sums of cubes of increments of [math]B[/math] converges in law to an independent Brownian motion as the size of the increments tends to zero. In joint work with Chris Burdzy and David Nualart, we study the convergence in law of two subsequences of [math]\{W_n\}[/math]. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.

## Thursday, October 25, Mihai Stoiciu, Williams College

Title: Random Matrices with Poisson Eigenvalue Statistics

Abstract: Several classes of random self-adjoint and random unitary matrices exhibit Poisson microscopic eigenvalue statistics. We will outline the general strategy for proving these results and discuss other models where the Poisson statistics is conjectured. We will also explain how changes in the distribution of the matrix coefficients produce changes in the microscopic eigenvalue distribution and give a transition from Poisson to the picket fence distribution.

## Thursday, November 8, Michael Kozdron, University of Regina

Title: TBA

Abstract: TBA

## Tuesday, November 27, Michael Damron, Princeton

Title: Busemann functions and infinite geodesics in first-passage percolation

Abstract: In first-passage percolation we study the chemical distance in the weighted graph Z^d, where the edge weights are given by a translation-ergodic (typically i.i.d.) distribution. A main open question is to describe the behavior of very long or infinite geodesics. In particular, one would like to know if there are infinite geodesics with asymptotic directions, how many are there, and if infinite geodesics in the same direction coalesce. Some of these questions were addressed in the late 90's by Newman and collaborators under strong assumptions on the limiting shape and weight distribution. I will discuss work with Jack Hanson (Ph. D. student at Princeton) where we develop a framework for working with distributional limits of Busemann functions and use them to prove a form of coalescence of geodesics constructed in any deterministic direction. We also prove existence of infinite geodesics which are asymptotically directed in sectors. Last, we introduce a purely directional condition which replaces Newman's global curvature condition and whose assumption implies the existence of directional geodesics.

## Thursday, December 6, Scott McKinley, University of Florida

Title: TBA

Abstract: TBA

## Thursday, December 13, Karl Liechty, University of Michigan

Title: TBA

Abstract: TBA