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 \[\bigl(\ kappa_{\beta}+o(1)\bigr)\lambda^{\gamma_{\beta}}\exp\biggl(-{\bet a}{64}\lambda^2+\biggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr)\] as $\lambda\to\infty$, where \[\gamma_{\beta}={1}{4}\biggl({\beta}{2}+{2}{\beta}-3\biggr)\] 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.