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Beta ensembles, stochastic Airy spectrum, and a diffusion

Authors: José A. Ramírez, Brian Rider and Bálint Virág
Journal: J. Amer. Math. Soc. 24 (2011), 919-944
MSC (2010): Primary 60F05, 60H25
Published electronically: May 6, 2011
MathSciNet review: 2813333
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Abstract: 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.

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Additional Information

José A. Ramírez
Affiliation: Department of Mathematics, Universidad de Costa Rica, San Jose 2060, Costa Rica

Brian Rider
Affiliation: Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, Colorado 80309

Bálint Virág
Affiliation: Department of Mathematics and Statistics, University of Toronto, Ontario, M5S 2E4, Canada
MR Author ID: 641409

Keywords: Random matrices, Tracy-Widom laws, random Schrödinger
Received by editor(s): November 3, 2009
Received by editor(s) in revised form: November 9, 2010
Published electronically: May 6, 2011
Additional Notes: The second author was supported in part by NSF grants DMS-0505680 and DMS-0645756.
The third author was supported in part by a Sloan Foundation fellowship, by the Canada Research Chair program, and by NSERC and Connaught research grants.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.