Large gaps between random eigenvalues

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

(κ_β+o(1))λ^γ_β exp((−β/64)λ²+(β/8−1/4)λ)

as λ→∞, where

γ_β=1/4(β/2+2/β−3)

and κ_β 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.