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}}expbigg
l(-{bet a}{64}lambda^2+biggl({beta}{8}-{1}{4}biggr)lambdabiggr)] as $lambdatoinfty$, where [gamma_{beta}={1}{4}biggl({beta}{2}+{2}{beta}-3biggr)] 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.
We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of
the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $beta$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2.
