Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

TL;DR

This paper explores the role of orthogonal and symplectic Harish-Chandra integrals in deriving eigenvalue distributions for real matrix product ensembles, connecting these results to known distributions of complex Gaussian matrix products.

Contribution

It provides explicit eigenvalue distributions for certain real matrix products using Harish-Chandra integrals and introduces a diffusion equation derivation for these integrals.

Findings

Eigenvalue distributions for specific real matrix products are derived.

Results connect real matrix product eigenvalues to complex Gaussian matrix singular values.

A diffusion equation approach for Harish-Chandra integrals is presented.

Abstract

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and particular Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure $X_M \cdots X_1( iA) X_1^T \cdots X_M^T$, where each $X_i$ is a standard real Gaussian matrix, and $A$ is a real anti-symmetric matrix can be determined. For $M=1$ and $A$ the bidiagonal anti-symmetric matrix with 1's above the diagonal, this reclaims results of Defosseux. For general $M$, and this choice of $A$, or $A$ itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.