Beta ensembles, stochastic Airy spectrum, and a diffusion

Ramírez, José ¹   ; Rider, Brian ²   ; Virág, Bálint ³

¹ Department of Mathematics, Universidad de Costa Rica, San Jose 2060, Costa Rica
² Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, Colorado 80309
³ Department of Mathematics and Statistics, University of Toronto, Ontario, M5S 2E4, Canada

Journal of the American Mathematical Society, Tome 24 (2011) no. 4, pp. 919-944

Résumé

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator $-\frac {d^2}{dx^2} + x + \frac {2}{\sqrt {\beta }} b_x^{\prime }$ restricted to the positive half-line, where $b_x^{\prime }$ is white noise. In doing so we extend the definition of the Tracy-Widom($\beta$) distributions to all $\beta >0$ and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

Détail

DOI : 10.1090/S0894-0347-2011-00703-0

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