Large gaps between random eigenvalues

Abstract

We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\ kappa_β+o(1)\bigr)λ^{γ_β}\exp\biggl(-{\bet a}{64}λ^2+\biggl(β{8}-{1}{4}\biggr)λ\biggr)\] as $\lambda \to \infty$, where \[γ_β={1}{4}\biggl(β{2}+{2}β-3\biggr)\] and ${\kappa }_{\beta }$ 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.

Published in at http://dx.doi.org/10.1214/09-AOP508 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Introduction to Random Matrices - Theory and Practice
• Continuum limits of random matrices and the Brownian carousel
• Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle

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• Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities
• Operator limits of random matrices
• The many faces of the stochastic zeta function
• Critical Fluctuations for the Spherical Sherrington-Kirkpatrick Model in an External Field
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• The Dyson and Coulomb games
• Limits for circular Jacobi beta-ensembles
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