## Beta ensembles, stochastic Airy spectrum, and a diffusion

HTML articles powered by AMS MathViewer

- 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
- PDF | 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.## References

- Jinho Baik, Percy Deift, and Kurt Johansson,
*On the distribution of the length of the longest increasing subsequence of random permutations*, J. Amer. Math. Soc.**12**(1999), no. 4, 1119–1178. MR**1682248**, DOI 10.1090/S0894-0347-99-00307-0 - Jinho Baik, Gérard Ben Arous, and Sandrine Péché,
*Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices*, Ann. Probab.**33**(2005), no. 5, 1643–1697. MR**2165575**, DOI 10.1214/009117905000000233 - Jinho Baik, Robert Buckingham, and Jeffery DiFranco,
*Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function*, Comm. Math. Phys.**280**(2008), no. 2, 463–497. MR**2395479**, DOI 10.1007/s00220-008-0433-5 - Yu. Baryshnikov,
*GUEs and queues*, Probab. Theory Related Fields**119**(2001), no. 2, 256–274. MR**1818248**, DOI 10.1007/PL00008760 - S. Cambronero and 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**, DOI 10.1002/(SICI)1097-0312(199910)52:10<1277::AID-CPA5>3.0.CO;2-L - Santiago Cambronero, B. Rider, and José Ramírez,
*On the shape of the ground state eigenvalue density of a random Hill’s equation*, Comm. Pure Appl. Math.**59**(2006), no. 7, 935–976. MR**2222441**, DOI 10.1002/cpa.20104 - P. A. Deift,
*Orthogonal polynomials and random matrices: a Riemann-Hilbert approach*, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR**1677884** - Percy Deift and Dimitri Gioev,
*Random matrix theory: invariant ensembles and universality*, Courant Lecture Notes in Mathematics, vol. 18, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2009. MR**2514781**, DOI 10.1090/cln/018 - Patrick Desrosiers and Peter J. Forrester,
*Hermite and Laguerre $\beta$-ensembles: asymptotic corrections to the eigenvalue density*, Nuclear Phys. B**743**(2006), no. 3, 307–332. MR**2227950**, DOI 10.1016/j.nuclphysb.2006.03.002 - Momar Dieng,
*Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations*, Int. Math. Res. Not.**37**(2005), 2263–2287. MR**2181265**, DOI 10.1155/IMRN.2005.2263 - I. Dumitriu,
*Personal communication*, 2006. - Ioana Dumitriu and Alan Edelman,
*Matrix models for beta ensembles*, J. Math. Phys.**43**(2002), no. 11, 5830–5847. MR**1936554**, DOI 10.1063/1.1507823 - L. Dumaz,
*The Tracy-Widom right tail*. Master’s thesis, Ecole Normale Supérieure, 2009. - L. Dumav, B. Virág
*The right tail exponent of the Tracy-Widom-beta distribution.*Preprint, arXiv:1102.4818 (2011). - Alan Edelman and Brian D. Sutton,
*From random matrices to stochastic operators*, J. Stat. Phys.**127**(2007), no. 6, 1121–1165. MR**2331033**, DOI 10.1007/s10955-006-9226-4 - 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). - Noureddine El Karoui,
*Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices*, Ann. Probab.**35**(2007), no. 2, 663–714. MR**2308592**, DOI 10.1214/009117906000000917 - Stewart N. Ethier and Thomas G. Kurtz,
*Markov processes*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR**838085**, DOI 10.1002/9780470316658 - Patrik L. Ferrari and Herbert Spohn,
*Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process*, Comm. Math. Phys.**265**(2006), no. 1, 1–44. MR**2217295**, DOI 10.1007/s00220-006-1549-0 - P. J. Forrester,
*Log-gases and random matrices*, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010. MR**2641363**, DOI 10.1515/9781400835416 - Bertrand I. Halperin,
*Green’s functions for a particle in a one-dimensional random potential*, Phys. Rev. (2)**139**(1965), A104–A117. MR**187859** - Kurt Johansson,
*Shape fluctuations and random matrices*, Comm. Math. Phys.**209**(2000), no. 2, 437–476. MR**1737991**, DOI 10.1007/s002200050027 - Iain M. Johnstone,
*On the distribution of the largest eigenvalue in principal components analysis*, Ann. Statist.**29**(2001), no. 2, 295–327. MR**1863961**, DOI 10.1214/aos/1009210544 - Rowan Killip and Mihai Stoiciu,
*Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles*, Duke Math. J.**146**(2009), no. 3, 361–399. MR**2484278**, DOI 10.1215/00127094-2009-001 - H. P. McKean,
*A limit law for the ground state of Hill’s equation*, J. Statist. Phys.**74**(1994), no. 5-6, 1227–1232. MR**1268791**, DOI 10.1007/BF02188225 - Michael Prähofer and Herbert Spohn,
*Scale invariance of the PNG droplet and the Airy process*, J. Statist. Phys.**108**(2002), no. 5-6, 1071–1106. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR**1933446**, DOI 10.1023/A:1019791415147 - José A. Ramírez and Brian Rider,
*Diffusion at the random matrix hard edge*, Comm. Math. Phys.**288**(2009), no. 3, 887–906. MR**2504858**, DOI 10.1007/s00220-008-0712-1 - Brian D. Sutton,
*The stochastic operator approach to random matrix theory*, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR**2717319** - Craig A. Tracy and Harold Widom,
*Level-spacing distributions and the Airy kernel*, Comm. Math. Phys.**159**(1994), no. 1, 151–174. MR**1257246**, DOI 10.1007/BF02100489 - Craig A. Tracy and Harold Widom,
*On orthogonal and symplectic matrix ensembles*, Comm. Math. Phys.**177**(1996), no. 3, 727–754. MR**1385083**, DOI 10.1007/BF02099545 - Hale F. Trotter,
*Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő*, Adv. in Math.**54**(1984), no. 1, 67–82. MR**761763**, DOI 10.1016/0001-8708(84)90037-9 - Benedek Valkó and Bálint Virág,
*Continuum limits of random matrices and the Brownian carousel*, Invent. Math.**177**(2009), no. 3, 463–508. MR**2534097**, DOI 10.1007/s00222-009-0180-z - Benedek Valkó and Bálint Virág,
*Large gaps between random eigenvalues*, Ann. Probab.**38**(2010), no. 3, 1263–1279. MR**2674999**, DOI 10.1214/09-AOP508

## 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