BROWNIAN PARTICLES WITH ELECTROSTATIC REPULSION ON THE CIRCLE: DYSON'S MODEL FOR UNITARY RANDOM MATRICES REVISITED

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 dene the nite 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 innity (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 ( ;) is the only limiting distribution of t when t goes to innity and t has an analytical density.

[1]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[2]  R. Pinsky.  ON THE CONVERGENCE OF DIFFUSION PROCESSES CONDITIONED TO REMAIN IN A BOUNDED REGION FOR LARGE TIME TO LIMITING POSITIVE RECURRENT DIFFUSION PROCESSES , 1985 .

[3]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[4]  Yasumasa Saisho,et al.  Stochastic differential equations for multi-dimensional domain with reflecting boundary , 1987 .

[5]  Emmanuel Cépa,et al.  Diffusing particles with electrostatic repulsion , 1997 .

[6]  H. McKean.  STOCHASTIC INTEGRAL EQUATIONS ( d = 1) , 1969 .

[7]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[8]  E. Cépa.  Problème de Skorohod multivoque , 1998 .

[9]  F. Schlögl.  D. Williams: Diffusion, Markov Processes, and Martingales, Vol. I: Foundations. John Wiley & Sons, Chichester, New York, Brisbane, Toronto 1979. 237 Seiten, Preis: £ 13.50 , 1980 .

[10]  G. Lawler,et al.  The geometry of the Brownian curve , 1993 .

[11]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[12]  Wendelin Werner,et al.  Non-Colliding Brownian Motions on the Circle , 1996 .

[13]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[14]  Emmanuel Cépa.  Équations différentielles stochastiques multivoques , 1994 .

[15]  François Bouchut,et al.  A NON-LINEAR STOCHASTIC DIFFERENTIAL EQUATION INVOLVING THE HILBERT TRANSFORM , 1999 .

[16]  David J. Grabiner.  Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices , 1997, math/9708207.

[17]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[18]  W. Feller.  Diffusion processes in one dimension , 1954 .

[19]  M. Métivier,et al.  Quelques problemes lies aux systemes infinis de particules et leurs limites , 1986 .

[20]  F. Dyson.  A Brownian‐Motion Model for the Eigenvalues of a Random Matrix , 1962 .

[21]  Pierre Bernard,et al.  Lectures on Probability Theory and Statistics: Ecole d'Ete de Probabilites de Saint-Flour XXVI - 1996 , 1997 .

[22]  L. Rogers,et al.  Interacting Brownian particles and the Wigner law , 1993 .

[23]  Hiroshi Tanaka.  Stochastic differential equations with reflecting boundary condition in convex regions , 1979 .

[24]  Hiroshi Tanaka,et al.  A diffusion process in a singular mean-drift-field , 1985 .

[25]  Terence Chan.  The Wigner semi-circle law and eigenvalues of matrix-valued diffusions , 1992 .