Riemannian symmetric superspaces and their origin in random matrix theory

TL;DR

This paper explores Gaussian random matrix ensembles linked to symmetric superspaces, revealing their role in mesoscopic physics and spectral correlation analysis through Riemannian superspace integrals.

Contribution

It introduces a new geometric framework connecting random matrix ensembles with Riemannian symmetric superspaces, advancing the understanding of spectral correlations in complex systems.

Findings

Spectral correlations are expressed as integrals over Riemannian symmetric superspaces.

Ensembles are linked to Cartan's symmetric spaces and supersymmetric geometry.

The approach unifies random matrix theory with supergeometry concepts.

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.