# Brownian motion and random matrices

December 14 to December 18, 2009

at the

American Institute of Mathematics, Palo Alto, California

organized by

Peter Forrester, Brian Rider, and Balint Virag

## Original Announcement

This workshop will be devoted to $\beta$-generalizations of the classical ensembles in random matrix theory. These are certain tridiagonal and unitary Hessenberg matrices, with an eigenvalue p.d.f. generalizing that of Gaussian Hermitian matrices and Haar distributed unitary matrices.

The $\beta$-generalizations lead to characterizations of the limiting eigenvalue distributions by way of certain stochastic differential equations involving Brownian motion. This in turn has been used to solve some previously intractable problems in random matrix theory, an example being the large distance asymptotic expansion of the spacing distributions for general $\beta$.

The time is ripe to use the stochastic characterization to tackle other problems fundamental to random matrix theory.

Universality: Do the bulk scaled eigenvalues in the $\beta$-generalized Gaussian and circular ensembles have the same distribution, and what if the Gaussian is replaced by say a quartic? Seemingly different stochastic descriptions apply in these cases, and the task is to show that they are in fact identical.

Phase transitions: Gaussian ensembles can be generalized to have Brownian motion valued entries, with one of the simplest initial conditions being to start all eigenvalues off at the origin except for one outlier. By tuning the value of the position of the outlier as a function of the size of the matrix, it is possible to get a critical regime, which is essentially the one studied in the context of spiked models. The problem here is to use the tridiagonal matrix models to study this setting, and to apply the findings to spiked models.

Integrability: Random matrix theory is a rich arena of integrability, with key probabilistic quantites known in terms of solutions of certain (non-stochastic) d.e.'s. One would like to use the s.d.e.'s, or other structures not restricted to the classical couplings, as a pathway to exact results for general $\beta$.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: