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

Emmanuel Cépa; Dominique Lépingle

ESAIM: Probability and Statistics (2010)

  • Volume: 5, page 203-224
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Cépa, Emmanuel, and Lépingle, Dominique. "Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited." ESAIM: Probability and Statistics 5 (2010): 203-224. <http://eudml.org/doc/197772>.

@article{Cépa2010,
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. },
author = {Cépa, Emmanuel, Lépingle, Dominique},
journal = {ESAIM: Probability and Statistics},
keywords = {Repulsive particles; multivalued stochastic differential equations; empirical measure process.; repulsive particles; deterministic second-order partial differential equations},
language = {eng},
month = {3},
pages = {203-224},
publisher = {EDP Sciences},
title = {Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited},
url = {http://eudml.org/doc/197772},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Cépa, Emmanuel
AU - Lépingle, Dominique
TI - Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 203
EP - 224
AB - 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.
LA - eng
KW - Repulsive particles; multivalued stochastic differential equations; empirical measure process.; repulsive particles; deterministic second-order partial differential equations
UR - http://eudml.org/doc/197772
ER -

References

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