Moments of normally distributed random matrices - Bijective explicit evaluation

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.