Riemannian symmetric superspaces and their origin in random matrix theory

Abstract

Gaussian random matrix ensembles defined over the tangent spaces of the large families of Cartan's symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics since they describe the universal ergodic limit of disordered and chaotic single particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,M_r) where G/H is a complex-analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and M_r is a maximal Riemannian submanifold of the support of G/H.