Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited

MAPMO, UMR 6628, bâtiment de Mathématiques, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; cepa@labomath.univ-orleans.fr. and

Received: 20 December 2000
Revised: 8 October 2001

Abstract

The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N x N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to define the finite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to infinity (through the empirical measure process). We prove that a limiting measure-valued process exists and is the unique solution of a deterministic second-order PDE. The uniform law on [-π;π] is the only limiting distribution of µ_t when t goes to infinity and µ_t has an analytical density.