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
as λ→∞, where
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.
"Large gaps between random eigenvalues." Ann. Probab. 38 (3) 1263 - 1279, May 2010. https://doi.org/10.1214/09-AOP508