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
Mots clés : 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},
     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, Tome 5 (2001), pp. 203-224. http://www.numdam.org/item/PS_2001__5__203_0/

[1] A. Bonami, F. Bouchut, E. Cépa and D. Lépingle, A nonlinear SDE involving Hilbert transform. J. Funct. Anal. 165 (1999) 390-406. | MR | Zbl

[2] E. Cépa, Équations différentielles stochastiques multivoques. Sémin. Probab. XXIX (1995) 86-107. | Numdam | MR | Zbl

[3] E. Cépa, Problème de Skorohod multivoque. Ann. Probab. 26 (1998) 500-532. | MR | Zbl

[4] E. Cépa and D. Lépingle, Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107 (1997) 429-449. | MR | Zbl

[5] T. Chan, The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probab. Theory Related Fields 93 (1992) 249-272. | MR | Zbl

[6] B. Duplantier, G.F. Lawler, J.F. Le Gall and T.J. Lyons, The geometry of Brownian curve. Bull. Sci. Math. 2 (1993) 91-106. | MR | Zbl

[7] F.J. Dyson, A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191-1198. | MR | Zbl

[8] W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 (1954) 1-31. | MR | Zbl

[9] D.J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré 35 (1999) 177-204. | Numdam | MR | Zbl

[10] D. Hobson and W. Werner, Non-colliding Brownian motion on the circle. Bull. London Math. Soc. 28 (1996) 643-650. | MR | Zbl

[11] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer, Berlin Heidelberg New York (1988). | MR | Zbl

[12] P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. | MR | Zbl

[13] H.P. Mckean, Stochastic integrals. Academic Press, New York (1969). | MR | Zbl

[14] M.L. Mehta, Random matrices. Academic Press, New York (1991). | MR | Zbl

[15] M. Metivier, Quelques problèmes liés aux systèmes infinis de particules et leurs limites. Sémin. Probab. XX (1986) 426-446. | Numdam | MR | Zbl

[16] M. Nagasawa and H. Tanaka, A diffusion process in a singular mean-drift field. Z. Wahrsch. Verw. Gebiete 68 (1985) 247-269. | MR | Zbl

[17] R.G. Pinsky, On the convergence of diffusion processes conditioned to remain in a bounded region for large times to limiting positive recurrent diffusion processes. Ann. Probab. 13 (1985) 363-378. | MR | Zbl

[18] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer Verlag, Berlin Heidelberg (1991). | MR | Zbl

[19] L.C.G. Rogers and Z. Shi, Interacting Brownian particles and the Wigner law. Probab. Theory Related Fields 95 (1993) 555-570. | MR | Zbl

[20] L.C.G. Rogers and D. Williams, Diffusions, Markov processes and Martingales. Wiley and Sons, New York (1987). | MR | Zbl

[21] Y. Saisho, Stochastic differential equations for multidimensional domains with reflecting boundary. Probab. Theory Related Fields 74 (1987) 455-477. | MR | Zbl

[22] H. Spohn, Dyson's model of interacting Brownian motions at arbitrary coupling strength. Markov Process. Related Fields 4 (1998) 649-661. | Zbl

[23] A.S. Sznitman, Topics in propagation of chaos. École d'été Probab. Saint-Flour XIX (1991) 167-251. | Zbl

[24] H. Tanaka, Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 (1979) 163-177. | MR | Zbl

[25] D. Voiculescu, Lectures on free probability theory. École d'été Probab. Saint-Flour (1998). | Zbl