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repulsive particles

ESAIM: Probability and Statistics (1)

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Brownian particles with electrostatic repulsionon the circle: Dyson's model for unitary random matrices revisited
Emmanuel Cépa, Dominique Lépingle
Journal:

ESAIM: Probability and Statistics / Volume 5 / 2001

Published online by Cambridge University Press:

15 August 2002, pp. 203-224

Print publication:

2001
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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 µt when t goes to infinity and µt has an analytical density.