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

Emmanuel Cépa; Dominique Lépingle

ESAIM: Probability and Statistics (2001)

- 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 (2001): 203-224. <http://eudml.org/doc/104274>.

@article{Cépa2001,

abstract = {The brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices $N \times N$ 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 $[- \pi ; \pi ]$ is the only limiting distribution of $\mu _t$ when $t$ goes to infinity and $\mu _t$ has an analytical density.},

author = {Cépa, Emmanuel, Lépingle, Dominique},

journal = {ESAIM: Probability and Statistics},

keywords = {repulsive particles; multivalued stochastic differential equations; empirical measure process; deterministic second-order partial differential equations},

language = {eng},

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/104274},

volume = {5},

year = {2001},

}

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

PY - 2001

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 $N \times N$ 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 $[- \pi ; \pi ]$ is the only limiting distribution of $\mu _t$ when $t$ goes to infinity and $\mu _t$ has an analytical density.

LA - eng

KW - repulsive particles; multivalued stochastic differential equations; empirical measure process; deterministic second-order partial differential equations

UR - http://eudml.org/doc/104274

ER -

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