Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

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 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 [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] 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.