Beta ensembles, stochastic Airy spectrum, and a diffusion
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- by José A. Ramírez, Brian Rider and Bálint Virág
- J. Amer. Math. Soc. 24 (2011), 919-944
- DOI: https://doi.org/10.1090/S0894-0347-2011-00703-0
- Published electronically: May 6, 2011
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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.References
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Bibliographic 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