Benedek Valkó

Random Matrices and the Brownian Carousel

Understanding the eigenvalue distribution of large random matrices is an
important problem for several branches of mathematics - probability
theory, number theory, combinatorics, mathematical physics, operator
algebras - and is expected to have far reaching applications.

It has been known since Wigner (since the 1950s) that for a large class
of
random matrix models the rescaled spectral density converges to the
semi-circle distribution. The finer asymptotics of the spectrum, i.e.
local limits of the eigenvalue process are still not fully understood,
although a lot of advance has been made for special models.

In this talk we describe the scaling limit of the spectrum of the
so-called beta-ensembles (which includes such much studied examples as
the
Gaussian orthogonal, unitary, or symplectic ensembles). The limiting
point process can be defined as a simple functional of Brownian motion
in
the hyperbolic plane - the 'Brownian Carousel'.

(joint work with Bálint Virág)