Brownian particles with electrostatic repulsion on the circle : Dyson's model for unitary random matrices revisited
ESAIM: Probability and Statistics, Tome 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 [-π;π] is the only limiting distribution of μ t when t goes to infinity and μ t has an analytical density.

Classification : 60K35, 60F05, 60H10, 60J60
Keywords: repulsive particles, multivalued stochastic differential equations, empirical measure process 
@article{PS_2001__5__203_0,
     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},
     year = {2001},
     publisher = {EDP Sciences},
     volume = {5},
     zbl = {1002.60093},
     language = {en},
     url = {https://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, Tome 5 (2001), pp. 203-224. https://www.numdam.org/item/PS_2001__5__203_0/

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