Volume 5, 2001
|Page(s)||203 - 224|
|Published online||15 August 2002|
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; email@example.com. and
Revised: 8 October 2001
The Brownian motion model introduced by Dyson  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.
Mathematics Subject Classification: 60K35 / 60F05 / 60H10 / 60J60
Key words: Repulsive particles / multivalued stochastic differential equations / empirical measure process.
© EDP Sciences, SMAI, 2001
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