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 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 · · · X 1 (iA)X T 1 · · · X T M , 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.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

“…where each Y i is a particular truncation of a Haar distributed complex unitary matrix, see Proposition 4.5. This observation compliments the one in [18] made for the Gaussian case.…”

“…Also, the integrand is independent of k, while the dependence on t 12 factorises. Integrating over these variables gives We remark that the result of Corollary 4.1 includes the result of [18,Corollary 4.3], which (after a minor change of notation) tells us that with X a 2N × 2n (N ≥ n) standard real Gaussian matrix, and A = diag(a 1 , . .…”

“…In a recent work involving two of the present authors [18] it has been shown that taking a viewpoint of Hermitised products does allow for a natural theory of explicit, structured formulas for eigenvalue jPDFs and associated correlation functions for the X i being real Gaussian matrices. For this to be possible, Eq.…”

“…(1.4) must first be generalise to the form X M · · · X 2 X 1 AX T 1 X T 2 · · · X T M , (1.5) where A is a real anti-symmetric matrix. The explicit results of [18] are based on the Harish-Chandra group integral for the orthogonal group [22,12], K=O(2n) (2k)! det[cosh x i y j ] i,j=1,...,n ∆ n (x 2 )∆ n (y 2 ) , (1.6) where X, Y are 2n × 2n antisymmetric matrices with singular values {x j }, {y j }, and its analogue for the odd dimensional case.…”

“…In Section 4 the general fomalism is specialised to the case that the X i in Eq. (1.5) are either real Gaussians, so reclaiming the corresponding results in [18], or truncations of real orthogonal matrices (Jacobi ensemble). In particular the case of the jPDF of the squared singular values of A = ı1 1 n ⊗ τ 2 (τ 2 is the second Pauli matrix) surprisingly coincides with the jPDF of the…”