Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited

Emmanuel Cépa; Dominique Lépingle

ESAIM: Probability and Statistics (2001)

  • Volume: 5, page 203-224
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Cépa, Emmanuel, and Lépingle, Dominique. "Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited." ESAIM: Probability and Statistics 5 (2001): 203-224. <http://eudml.org/doc/104274>.

@article{Cépa2001,
abstract = {The brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices $N \times N$ 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 $[- \pi ; \pi ]$ is the only limiting distribution of $\mu _t$ when $t$ goes to infinity and $\mu _t$ has an analytical density.},
author = {Cépa, Emmanuel, Lépingle, Dominique},
journal = {ESAIM: Probability and Statistics},
keywords = {repulsive particles; multivalued stochastic differential equations; empirical measure process; deterministic second-order partial differential equations},
language = {eng},
pages = {203-224},
publisher = {EDP-Sciences},
title = {Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited},
url = {http://eudml.org/doc/104274},
volume = {5},
year = {2001},
}

TY - JOUR
AU - Cépa, Emmanuel
AU - Lépingle, Dominique
TI - Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited
JO - ESAIM: Probability and Statistics
PY - 2001
PB - EDP-Sciences
VL - 5
SP - 203
EP - 224
AB - The brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices $N \times N$ 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 $[- \pi ; \pi ]$ is the only limiting distribution of $\mu _t$ when $t$ goes to infinity and $\mu _t$ has an analytical density.
LA - eng
KW - repulsive particles; multivalued stochastic differential equations; empirical measure process; deterministic second-order partial differential equations
UR - http://eudml.org/doc/104274
ER -

References

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  1. [1] A. Bonami, F. Bouchut, E. Cépa and D. Lépingle, A nonlinear SDE involving Hilbert transform. J. Funct. Anal. 165 (1999) 390-406. Zbl0935.60095MR1698948
  2. [2] E. Cépa, Équations différentielles stochastiques multivoques. Sémin. Probab. XXIX (1995) 86-107. Zbl0833.60079MR1459451
  3. [3] E. Cépa, Problème de Skorohod multivoque. Ann. Probab. 26 (1998) 500-532. Zbl0937.34046MR1626174
  4. [4] E. Cépa and D. Lépingle, Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107 (1997) 429-449. Zbl0883.60089MR1440140
  5. [5] T. Chan, The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probab. Theory Related Fields 93 (1992) 249-272. Zbl0767.60050MR1176727
  6. [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. Zbl0778.60058MR1205413
  7. [7] F.J. Dyson, A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191-1198. Zbl0111.32703MR148397
  8. [8] W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 (1954) 1-31. Zbl0059.11601MR63607
  9. [9] D.J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré 35 (1999) 177-204. Zbl0937.60075MR1678525
  10. [10] D. Hobson and W. Werner, Non-colliding Brownian motion on the circle. Bull. London Math. Soc. 28 (1996) 643-650. Zbl0853.60060MR1405497
  11. [11] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer, Berlin Heidelberg New York (1988). Zbl0638.60065MR917065
  12. [12] P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. Zbl0598.60060MR745330
  13. [13] H.P. McKean, Stochastic integrals. Academic Press, New York (1969). Zbl0191.46603MR247684
  14. [14] M.L. Mehta, Random matrices. Academic Press, New York (1991). Zbl0780.60014MR1083764
  15. [15] M. Metivier, Quelques problèmes liés aux systèmes infinis de particules et leurs limites. Sémin. Probab. XX (1986) 426-446. Zbl0622.60112MR942037
  16. [16] M. Nagasawa and H. Tanaka, A diffusion process in a singular mean-drift field. Z. Wahrsch. Verw. Gebiete 68 (1985) 247-269. Zbl0558.60060MR771466
  17. [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. Zbl0567.60076MR781410
  18. [18] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer Verlag, Berlin Heidelberg (1991). Zbl0731.60002MR1083357
  19. [19] L.C.G. Rogers and Z. Shi, Interacting Brownian particles and the Wigner law. Probab. Theory Related Fields 95 (1993) 555-570. Zbl0794.60100MR1217451
  20. [20] L.C.G. Rogers and D. Williams, Diffusions, Markov processes and Martingales. Wiley and Sons, New York (1987). Zbl0826.60002MR921238
  21. [21] Y. Saisho, Stochastic differential equations for multidimensional domains with reflecting boundary. Probab. Theory Related Fields 74 (1987) 455-477. Zbl0591.60049MR873889
  22. [22] H. Spohn, Dyson’s model of interacting Brownian motions at arbitrary coupling strength. Markov Process. Related Fields 4 (1998) 649-661. Zbl0927.60086
  23. [23] A.S. Sznitman, Topics in propagation of chaos. École d’été Probab. Saint-Flour XIX (1991) 167-251. Zbl0732.60114
  24. [24] H. Tanaka, Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 (1979) 163-177. Zbl0423.60055MR529332
  25. [25] D. Voiculescu, Lectures on free probability theory. École d’été Probab. Saint-Flour (1998). Zbl1015.46037

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