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

The brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices $N\times N$ 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 $[-\pi ;\pi ]$ is the only limiting distribution of ${\mu }_t$ when $t$ goes to infinity and ${\mu }_t$ has an analytical density.