Universality for orthogonal and symplectic Laguerre-type ensembles

Percy Deift, D. Gioev, T. Kriecherbauer, M. Vanlessen

Research output: Contribution to journalArticle

Abstract

We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K n, β, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β=2) Laguerre-type ensembles have been proved by the fourth author in Ref. 23. The varying weight case at the hard spectral edge was analyzed in Ref. 13 for β=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in Refs. 7, 8 analogous results for Hermite-type ensembles. As in Refs. 7, 8 we use the version of the orthogonal polynomial method presented in Refs. 22, 25, to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from Ref. 23.

Original languageEnglish (US)
Pages (from-to)949-1053
Number of pages105
JournalJournal of Statistical Physics
Volume129
Issue number5-6
DOIs
StatePublished - Oct 2007

Fingerprint

Universality
polynomials
Ensemble
eigenvalues
Orthogonal Polynomials
Gap Function
Polynomial Methods
theorems
Smallest Eigenvalue
Largest Eigenvalue
statistics
Hermite
Random Matrices
Corollary
kernel
Statistics
Eigenvalue
Necessary
Theorem

Keywords

  • Bulk
  • Hard edge
  • Laguerre-type weights
  • Orthogonal and symplectic ensembles
  • Random matrix theory
  • Soft edge
  • Universality

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Universality for orthogonal and symplectic Laguerre-type ensembles. / Deift, Percy; Gioev, D.; Kriecherbauer, T.; Vanlessen, M.

In: Journal of Statistical Physics, Vol. 129, No. 5-6, 10.2007, p. 949-1053.

Research output: Contribution to journalArticle

Deift, Percy ; Gioev, D. ; Kriecherbauer, T. ; Vanlessen, M. / Universality for orthogonal and symplectic Laguerre-type ensembles. In: Journal of Statistical Physics. 2007 ; Vol. 129, No. 5-6. pp. 949-1053.
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