Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

TL;DR

This paper demonstrates that Riemannian Gaussian distributions on symmetric spaces are linked to random matrix ensembles, enabling analytical computation of marginals and revealing their connection to diffusion kernels and Brownian motions.

Contribution

It establishes the random matrix nature of Riemannian Gaussian distributions and develops methods for analytical marginal computation across various symmetric spaces.

Findings

Analytical marginals for Hermitian matrix distributions using orthogonal polynomials.

Efficient computation of marginals for positive definite matrices via Pfaffians.

Connection of distributions to diffusion kernels and non-intersecting Brownian motions.

Abstract

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of $m \times m$ symmetric positive definite matrices, we show how to efficiently compute by evaluating Pfaffians at specific values of $m$. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.