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Brownian particles with electrostatic repulsion on the circle: Dyson’s model for unitary random matrices revisited. (English) Zbl 1002.60093
The authors extend a Brownian motion model introduced by F. J. Dyson [J. Math. Phys. 3, 1191-1198 (1962; Zbl 0111.32703)] for the eigenvalues of unitary random matrices \(N\times N\) which 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, they study the behaviour of this system when the number of particles \(N\) goes to infinity (through the empirical measure process). Furthermore it is shown 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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