Brownian particles with electrostatic repulsion on the circle : Dyson's model for unitary random matrices revisited
ESAIM: Probability and Statistics, Volume 5 (2001), pp. 203-224.

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

Classification: 60K35,  60F05,  60H10,  60J60
Keywords: repulsive particles, multivalued stochastic differential equations, empirical measure process
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author = {C\'epa, Emmanuel and L\'epingle, Dominique},
title = {Brownian particles with electrostatic repulsion on the circle : {Dyson's} model for unitary random matrices revisited},
journal = {ESAIM: Probability and Statistics},
pages = {203--224},
publisher = {EDP-Sciences},
volume = {5},
year = {2001},
zbl = {1002.60093},
language = {en},
url = {http://www.numdam.org/item/PS_2001__5__203_0/}
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Cépa, Emmanuel; Lépingle, Dominique. Brownian particles with electrostatic repulsion on the circle : Dyson's model for unitary random matrices revisited. ESAIM: Probability and Statistics, Volume 5 (2001), pp. 203-224. http://www.numdam.org/item/PS_2001__5__203_0/

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