Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
This work provides a theoretical framework for understanding and computing properties of Riemannian Gaussian distributions, which is significant for researchers in mathematical physics and random matrix theory, offering new analytical tools for these distributions.
This paper demonstrates that Riemannian Gaussian distributions on symmetric spaces are equivalent to standard random matrix ensembles. This equivalence allows for the analytical computation of marginals of probability density functions, specifically for Hermitian matrices using Stieltjes-Wigert polynomials and for symmetric positive definite matrices by evaluating Pfaffians or constructing skew Stieltjes-Wigert polynomials.
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.