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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Beta ensembles, stochastic Airy spectrum, and a diffusion
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by José A. Ramírez, Brian Rider and Bálint Virág PDF
J. Amer. Math. Soc. 24 (2011), 919-944 Request permission

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
  • Email: alexander.ramirez_g@ucr.ac.cr
  • Brian Rider
  • Affiliation: Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, Colorado 80309
  • Email: brian.rider@colorado.edu
  • Bálint Virág
  • Affiliation: Department of Mathematics and Statistics, University of Toronto, Ontario, M5S 2E4, Canada
  • MR Author ID: 641409
  • Email: balint@math.toronto.edu
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 24 (2011), 919-944
  • MSC (2010): Primary 60F05, 60H25
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00703-0
  • MathSciNet review: 2813333