A diffusive matrix model for invariant $\beta$-ensembles

Romain Allez (University Paris Dauphine)
Alice Guionnet (ÉNS Lyon)


We define a new diffusive matrix model converging towards the $\beta$-Dyson Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction of $\beta$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $\beta< 1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues.

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Pages: 1-30

Publication Date: June 21, 2013

DOI: 10.1214/EJP.v18-2073


  • R. Allez, J.-P. Bouchaud and A. Guionnet, Invariant β-ensembles and the Gauss-Wigner crossover, Phys. Rev. Lett. 109, 094102 (2012).
  • Allez, Romain; Bouchaud, Jean-Philippe; Majumdar, Satya N.; Vivo, Pierpaolo. Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marčenko-Pastur law. J. Phys. A 46 (2013), no. 1, 015001, 22 pp. MR3001575
  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396
  • Cépa, Emmanuel; Lépingle, Dominique. Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107 (1997), no. 4, 429--449. MR1440140
  • Cépa, Emmanuel; Lépingle, Dominique. No multiple collisions for mutually repelling Brownian particles. Séminaire de Probabilités XL, 241--246, Lecture Notes in Math., 1899, Springer, Berlin, 2007. MR2409009
  • Dumitriu, Ioana; Edelman, Alan. Matrix models for beta ensembles. J. Math. Phys. 43 (2002), no. 11, 5830--5847. MR1936554
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940
  • Mehta, Madan Lal. Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991. xviii+562 pp. ISBN: 0-12-488051-7 MR1083764
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357

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