# Spring 2015

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

Title: TBA

## Thursday, November 6, Vadim Gorin, MIT

Title: Multilevel Dyson Brownian Motion and its edge limits.

Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of random Hermitian matrices on the other side. In my talk I will explain some reasons for this connection between two seemingly unrelated classes of stochastic systems, and how this relation can be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion will be the central object in the discussion.

(Based on joint papers with Misha Shkolnikov.)

## Friday, November 7, Tim Chumley, Iowa State University

Title: Random billiards and diffusion

Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system. The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.

## Thursday, November 13, Timo Seppäläinen, UW-Madison

Title: Variational formulas for directed polymer and percolation models

Abstract: Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.

## Monday, December 1, Joe Neeman, UT-Austin, 4pm, Room B239 Van Vleck Hall

Please note the unusual time and room.

Title: Some phase transitions in the stochastic block model

Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.

## Thursday, December 4, Arjun Krishnan, Fields Institute

Title: Variational formula for the time-constant of first-passage percolation

Abstract: Consider first-passage percolation with positive, stationary-ergodic weights on the square lattice in d-dimensions. Let $\displaystyle{ T(x) }$ be the first-passage time from the origin to $\displaystyle{ x }$ in $\displaystyle{ Z^d }$. The convergence of $\displaystyle{ T([nx])/n }$ to the time constant as $\displaystyle{ n }$ tends to infinity is a consequence of the subadditive ergodic theorem. This convergence can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula (duality principle) for the time-constant. Under a symmetry assumption, we will use the variational formula to construct an explicit iteration that produces the limit shape.

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