Genus expansion for real Wishart matrices

TL;DR

This paper derives an exact Euler characteristic expansion for moments and cumulants of real Wishart matrices, revealing their asymptotic behavior and leading to a central limit theorem in the large matrix limit.

Contribution

It introduces a genus expansion approach for real Wishart matrices, connecting topological and algebraic methods to analyze their asymptotic distributions.

Findings

Exact formula for moments and cumulants using Euler characteristic expansion

Identification of dominant terms in large matrix limit as spheres or collections of spheres

Establishment of a central limit theorem for asymptotic distribution

Abstract

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central limit-type theorem for the asymptotic distribution around the expected value.