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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
DOI: https://doi.org/10.1090/S0894-0347-2011-00703-0
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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  • 1. J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), 1119-1178. MR 1682248 (2000e:05006)
  • 2. J. Baik, G. Ben Arous, S. Péché, Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33 (2005), 1643-1697. MR 2165575 (2006g:15046)
  • 3. J. Baik, R. Buckingham, J. DiFranco, Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Comm. Math. Phys., 280 (2008), 463-497. MR 2395479 (2009e:33068)
  • 4. Y. Baryshnikov, GUEs and queues. Probab. Theory Rel. Fields. 119 (2001), 256-274. MR 1818248 (2002a:60165)
  • 5. S. Cambronero, H.P. McKean, The Ground State Eigenvalue of Hill's Equation with White Noise Potential. Comm. Pure Appl. Math. 52 (1999), no 10, 1277-1294. MR 1699969 (2001g:60158)
  • 6. S. Cambronero, J. Ramírez, B. Rider, On the shape of the ground state eigenvalue density of a random Hill's equation. Comm. Pure Appl Math. 59 (2006), 935-976. MR 2222441 (2007a:34036a)
  • 7. P. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics 3, AMS Providence, RI, 1999. MR 1677884 (2000g:47048)
  • 8. P. Deift, D. Gioev, Random Matrix Theory: invariant ensembles and universality. Courant Lecture Notes in Mathematics 18, AMS Providence, RI, 2009. MR 2514781 (2011f:60008)
  • 9. P. Desrosiers, P. Forrester, Hermite and Laguerre $ \beta$-ensembles: asymptotic corrections to the eigenvalue density. Nuc. Phys. B. 743 (2006), 307-332. MR 2227950 (2007f:82043)
  • 10. M. Dieng, Distribution functions for edge eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé representations. Int. Math. Res. Notices. 2005, 2263-2287. MR 2181265 (2006h:60014)
  • 11. I. Dumitriu, Personal communication, 2006.
  • 12. I. Dumitriu, A. Edelman, Matrix models for beta ensembles. J. Math. Phys. 43 (2002), 5830-5847. MR 1936554 (2004g:82044)
  • 13. L. Dumaz, The Tracy-Widom right tail. Master's thesis, Ecole Normale Supérieure, 2009.
  • 14. L. Dumav, B. Virág The right tail exponent of the Tracy-Widom-beta distribution. Preprint, arXiv:1102.4818 (2011).
  • 15. A. Edelman, B. Sutton, From random matrices to stochastic operators. J. Stat. Phys. 127 (2007), 1121-1165. MR 2331033 (2009b:82037)
  • 16. N. El Karoui, On the largest eigenvalue of Wishart matrices with identity covariance when $ n$, $ p$, and $ p/n \rightarrow \infty$. Preprint, arXiv:math.ST/0309355 (2003).
  • 17. N. El Karoui, Tracy-Widom limit for the largest eigenvalue of a large class of complex Wishart matrices. Ann. Probab. 35 (2007), 663-714. MR 2308592 (2007m:60057)
  • 18. S. Ethier, T. Kurtz, Markov processes, characterization and convergence, Wiley, 1986. MR 838085 (88a:60130)
  • 19. P. Ferrari, H. Spohn, Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 (2006), 1-44. MR 2217295 (2007g:82038a)
  • 20. P. Forrester, Log-Gases and Random Matrices. London Math. Soc. Monographs, Princeton University Press, 2010. MR 2641363 (2011d:82001)
  • 21. B.I. Halperin, Green's functions for a particle in a one-dimensional random potential. Phys. Rev. (2) 139 (1965), A104-A117. MR 0187859 (32:5304)
  • 22. K. Johansson, Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), 437-476. MR 1737991 (2001h:60177)
  • 23. I.M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001), 295-327. MR 1863961 (2002i:62115)
  • 24. S. Killip, M. Stoiciu, Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Random Matrix Ensembles. Duke Math. Journal 146 (2009), 361-399. MR 2484278 (2009k:81087)
  • 25. H.P. McKean, A limit law for the ground state of Hill's equation. J. Stat. Phys. 74 (1994), 1227-1232. MR 1268791 (95d:82036)
  • 26. M. Prähofer, H. Spohn, Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (2002), 1071-1106. MR 1933446 (2003i:82050)
  • 27. J. Ramírez, B. Rider, Diffusion at the random matrix hard edge. Comm. Math. Phys. 288 (2009), 887-906 MR 2504858 (2010g:47083)
  • 28. B.D. Sutton, The stochastic operator approach to random matrix theory. Ph.D. thesis, MIT, Department of Mathematics, 2005. MR 2717319
  • 29. C. Tracy, H. Widom, Level spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994), 151-174. MR 1257246 (95e:82003)
  • 30. C. Tracy, H. Widom On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (1996), 727-754. MR 1385083 (97a:82055)
  • 31. H. F. Trotter,(1984) Eigenvalue distributions of large Hermitian matrices; Wigner's semicircle law and a theorem of Kac, Murdock, and Szegő. Adv. in Math. 54 (1984), 67-82. MR 761763 (86c:60055)
  • 32. B. Valko, B. Virág, Continuum limits of random matrices and the Brownian carousel. Inventiones 177 (2009), 463-508. MR 2534097 (2011d:60023)
  • 33. B. Valko, B. Virág, Large gaps between random eigenvalues. Ann. Probab. 38 (2010), 1263-1279. MR 2674999

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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
Email: balint@math.toronto.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00703-0
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.

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