Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Understand

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.