# 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

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topCépa, 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 -

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