Matrix averages relating to the Ginibre ensembles

TL;DR

This paper uses zonal polynomials to compute matrix averages in Ginibre ensembles, generalizing previous results and deriving moments and eigenvalue sum formulas for real, complex, and quaternion cases.

Contribution

It extends the computation of matrix averages in Ginibre ensembles using zonal polynomials, covering real, complex, and quaternion cases with new formulas.

Findings

Derived formulas for moments of Ginibre ensembles in terms of hypergeometric functions.

Expressed eigenvalue power sums as finite sums of zonal polynomials.

Generalized previous results to broader classes of Ginibre ensembles.

Abstract

The theory of zonal polynomials is used to compute the average of a Schur polynomial of argument $AX$, where $A$ is a fixed matrix and $X$ is from the real Ginibre ensemble. This generalizes a recent result of Sommers and Khorozhenko [J. Phys. A {\bf 42} (2009), 222002], and furthermore allows analogous results to be obtained for the complex and real quaternion Ginibre ensembles. As applications, the positive integer moments of the general variance Ginibre ensembles are computed in terms of generalized hypergeometric functions, these are written in terms of averages over matrices of the same size as the moment to give duality formulas, and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials.