Wigner surmise for mixed symmetry classes in random matrix theory

Abstract

We consider the nearest-neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes, or between integrable and non-integrable systems. We derive analytical formulas for the spacing distributions of 2x2 or 4x4 matrices and show numerically that they provide very good approximations for those of random matrices with large dimension. This generalizes the Wigner surmise, which is valid for pure ensembles that are recovered as limits of the mixed ensembles. We show how the coupling parameters of small and large matrices must be matched depending on the local eigenvalue density.