# Fall 2014

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

## Thursday, September 11, Van Vleck B105, Melanie Matchett Wood, UW-Madison

Title: The distribution of sandpile groups of random graphs

Abstract:
The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture.

## Thursday, September 18, Jonathon Peterson, Purdue University

Title: Hydrodynamic limits for directed traps and systems of independent RWRE

Abstract:

We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed $\displaystyle{ v_0 \neq 0) }$. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not average out and so the specific instance of the environment chosen actually matters.

The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of directed traps. This talk is based on joint work with Milton Jara.

## Thursday, September 25, Sean O'Rourke, University of Colorado Boulder

Title: Singular values and vectors under random perturbation

Abstract: Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?

Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank. This talk is based on joint work with Van Vu and Ke Wang.

## Thursday, October 2, Jun Yin, UW-Madison

Title: Anisotropic local laws for random matrices

Abstract: In this talk, we introduce a new method of deriving local laws of random matrices. As applications, we will show the local laws and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix), and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.

## Thursday, October 9, No seminar due to Midwest Probability Colloquium

No seminar due to Midwest Probability Colloquium.

## Thursday, October 16, Firas Rassoul-Agha, University of Utah

Title: The growth model: Busemann functions, shape, geodesics, and other stories

Abstract: We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen.

## 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.