# "Continuum limits for beta ensembles"

## Speaker:

## Institution:

## Time:

## Location:

The beta ensembles of random matrix theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, and 4. We prove that the extremal eigenvalues for the general ensembles have limit laws described by the low lying spectrum of certain raandom Schroedinger operators, as conjectured by Edelman-Sutton. As a corollary, a second characterization of these laws is made the explosion probability of a simple one-dimensional diffusion. A complementary pictures is developed for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

# On the zeroes of random polynomials.

## Speaker:

## Institution:

## Time:

## Location:

# On the spectrum of large random reversible stochastic matrice

## Speaker:

## Institution:

## Time:

## Location:

We consider random matrices associated to random walks on the complete

graph with random weights. When the weights have finite second moment we

find Wigner-like behavior for the empirical spectral density. If the

weights have finite fourth moment we prove convergence of extremal

eigenvalues to the edge of the semi-circle law. The case of weights with

infinite second moment is also considered. In this case we prove

convergence of the spectral density on a suitable scale and the limiting

measure is characterized in terms certain Poisson weighted infinite

trees associated to the starting graph. Connections with recent work on

random matrices with i.i.d. heavy-tailed entries and several open

problems are also discussed. This is recent work in collaboration with

D. Chafai and C. Bordenave (from Univ. P.Sabatier, Toulouse - France).

# Heat kernel estimates for Dirichlet fractional Laplacian

## Speaker:

## Institution:

## Time:

## Location:

In this talk, we discuss the sharp two-sided estimates for the heat kernel of Dirichlet fractional Laplacian in open sets. This heat kernel is also the transition density of a rotationally symmetric -stable process killed upon leaving an open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets. This is a joint work with Zhen-Qing Chen and Renming Song.

# On the almost-sure invariance principle for random walk in random environment

## Speaker:

## Institution:

## Time:

## Location:

Consider a crystal formed of two types of atoms placed at the nodes of the

integer lattice. The type of each atom is chosen at random, but the crystal

is statistically shift-invariant. Consider next an electron hopping from atom

to atom. This electron performs a random walk on the integer lattice with

randomly chosen transition probabilities (since the configuration seen by

the electron is different at each lattice site). This process is highly

non-Markovian, due to the interaction between the walk and the

environment.

We will present a martingale approach to proving the invariance principle

(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains

and show how this can be transferred to a result for the above process

(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.