Large gaps between random eigenvalues
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 \[\bigl(\ kappa_β+o(1)\bigr)λ^{γ_β}\exp\biggl(-{\bet a}{64}λ^2+\biggl(β{8}-{1}{4}\biggr)λ\biggr)\] as $λ\to\infty$, where \[γ_β={1}{4}\biggl(β{2}+{2}β-3\biggr)\] 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.
math.PR↗