Spacings in Orthogonal and Symplectic Random Matrix Ensembles

It is one of the famous features of random matrices that several local eigenvalue statistics are for growing matrix sizes asymptotically described by a universal limiting law. It is universal in the sense that it does not depend on the details of the underlying matrix ensemble but on its symmetry class only. In this paper, we consider the universality of the nearest neighbor eigenvalue spacing distribution in invariant random matrix ensembles. Focusing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution converges in a uniform way. More precisely, the main result states that the expected Kolmogorov distance of the empirical spacing distribution from its universal limit converges to zero as the matrix size tends to infinity.