2017/11/29 by Peter J. Forrester, Jesper R. Ipsen, Forrester, Peter J. +6
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1711.10691
38 pages
arxiv created 2017/11/29 · openalex publication_date 2017/11/29 · arxiv updated 2017/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 XM ⋯ X1( iA) X1T ⋯ XMT, where each Xi 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.