Moments of normally distributed random matrices - Bijective explicit evaluation

TL;DR

This paper derives explicit formulas for the moments of eigenvalue distributions of certain random matrices, using combinatorial and symmetric function techniques, extending previous work to complex-valued matrices.

Contribution

It provides a bijective combinatorial method to evaluate moments of eigenvalues for random matrices involving symmetric and hermitian matrices, connecting hypermap enumeration with symmetric functions.

Findings

Explicit evaluation of moments in terms of monomial symmetric functions

Bijective correspondence between hypermaps and decorated forests

Extension of results to complex-valued matrices with hermitian matrices

Abstract

This paper is devoted to the distribution of the eigenvalues of $XUYU^t$ where $X$ and $Y$ are given symmetric matrices and $U$ is a random real valued square matrix of standard normal distribution. More specifically we look at its moments, i.e. the mathematical expectation of the trace of $(XUYU^t)^n$ for arbitrary integer $n$. Hanlon, Stanley, Stembridge (1992) showed that this quantity can be expressed in terms of some generating series for the connection coefficients of the double cosets of the hyperoctahedral group with the eigenvalues of $X$ and $Y$ as indeterminate. We provide an explicit evaluation of these series in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some decorated forests. As a corollary we provide a simple explicit evaluation of the moments of $XUYU^*$ when $U$ is complex valued and $X$ and $Y$ are given hermitian matrices.