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

Published online by Cambridge University Press:  15 August 2002

Emmanuel Cépa  and
Dominique Lépingle
Emmanuel Cépa
Affiliation:
MAPMO, UMR 6628, bâtiment de Mathématiques, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; cepa@labomath.univ-orleans.fr. and
Dominique Lépingle
Affiliation:
MAPMO, UMR 6628, bâtiment de Mathématiques, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; cepa@labomath.univ-orleans.fr. and
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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 µt when t goes to infinity and µt has an analytical density.

Keywords

Repulsive particlesmultivalued stochastic differential equationsempirical measure process.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 203 - 224
Copyright
© EDP Sciences, SMAI, 2001

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