Riemannian symmetric superspaces and their origin in random matrix theory

TL;DR

This work shows that Gaussian random‑matrix ensembles defined on the tangent spaces of Cartan’s symmetric spaces can be analyzed in the large‑$N$ limit by reducing spectral correlations to integrals over Riemannian symmetric superspaces. By developing a robust Berezin integration framework, Grassmann‑analytic continuation, and careful domain choices, the author derives a saddle‑point reduction to a single dominant supermanifold, ${ m Osp}(2n|2n)/{ m Gl}(n|n)$, with integration over a product of noncompact and compact sectors. The analysis extends the supersymmetry method beyond the standard Wigner–Dyson classes to ten symmetry families, locating precise geometric correspondences between random‑matrix ensembles and large families of symmetric superspaces (e.g., $M_B={ m SO}^*(2n)/{ m U}(n)$, $M_F={ m Sp}(n)/{ m U}(n)$). These results unify spectral correlation problems across mesoscopic normal‑superconducting hybrids and related disordered/chaotic systems, and set the stage for extensions to diffusion, localization, and transport phenomena in a geometric superspace framework.

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.