Translator Disclaimer
May 2010 Large gaps between random eigenvalues
Benedek Valkó, Bálint Virág
Ann. Probab. 38(3): 1263-1279 (May 2010). DOI: 10.1214/09-AOP508

Abstract

We show that in the point process limit of the bulk eigenvalues of β-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size λ is given by

(κβ+o(1))λγβ exp((−β/64)λ2+(β/8−1/4)λ)

as λ→∞, where

γβ=1/4(β/2+2/β−3)

and κβ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157–165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron–Martin–Girsanov transformation in stochastic calculus.

Citation

Download Citation

Benedek Valkó. Bálint Virág. "Large gaps between random eigenvalues." Ann. Probab. 38 (3) 1263 - 1279, May 2010. https://doi.org/10.1214/09-AOP508

Information

Published: May 2010
First available in Project Euclid: 2 June 2010

zbMATH: 1223.60009
MathSciNet: MR2674999
Digital Object Identifier: 10.1214/09-AOP508

Subjects:
Primary: 15B52, 60F10

Rights: Copyright © 2010 Institute of Mathematical Statistics

JOURNAL ARTICLE
17 PAGES


SHARE
Vol.38 • No. 3 • May 2010
Back to Top