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)