Universality limits for entire functions
HTML articles powered by AMS MathViewer
- by Mishko Mitkovski PDF
- Proc. Amer. Math. Soc. 141 (2013), 3119-3124 Request permission
Abstract:
Various statements on the distribution of eigenvalues of random matrices are obtained by considering the limiting behavior of the reproducing kernels of a certain naturally associated sequence of orthogonal polynomials. We establish a universal limiting behavior of this type in the case when the underlying measure does not have finite moments. In this case the orthogonal polynomials are replaced by a nested family of de Branges spaces of entire functions.References
- P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights, Internat. Math. Res. Notices 16 (1997), 759–782. MR 1472344, DOI 10.1155/S1073792897000500
- P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), no. 12, 1491–1552. MR 1711036, DOI 10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R
- P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
- M. G. Kreĭn, On a basic approximation problem of the theory of extrapolation and filtration of stationary random processes, Doklady Akad. Nauk SSSR (N.S.) 94 (1954), 13–16 (Russian). MR 0062980
- Eli Levin and Doron S. Lubinsky, Universality limits in the bulk for varying measures, Adv. Math. 219 (2008), no. 3, 743–779. MR 2442052, DOI 10.1016/j.aim.2008.06.010
- Doron S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math. (2) 170 (2009), no. 2, 915–939. MR 2552113, DOI 10.4007/annals.2009.170.915
- D. S. Lubinsky, Universality limits in the bulk for arbitrary measures on compact sets, J. Anal. Math. 106 (2008), 373–394. MR 2448991, DOI 10.1007/s11854-008-0053-1
- L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), no. 1-2, 109–147. MR 1435193, DOI 10.1007/BF02180200
- Barry Simon, Two extensions of Lubinsky’s universality theorem, J. Anal. Math. 105 (2008), 345–362. MR 2438429, DOI 10.1007/s11854-008-0039-z
- Vilmos Totik, Universality and fine zero spacing on general sets, Ark. Mat. 47 (2009), no. 2, 361–391. MR 2529707, DOI 10.1007/s11512-008-0071-3
Additional Information
- Mishko Mitkovski
- Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
- Address at time of publication: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
- MR Author ID: 898865
- Email: mishko@math.gatech.edu, mmitkov@clemson.edu
- Received by editor(s): November 14, 2011
- Published electronically: May 7, 2013
- Additional Notes: The author was supported in part by NSF grants #DMS-1001098 and #DMS-1101251.
- Communicated by: Richard Rochberg
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3119-3124
- MSC (2010): Primary 30D20
- DOI: https://doi.org/10.1090/S0002-9939-2013-11585-6
- MathSciNet review: 3068965