Gaussian distributions on Riemannian symmetric spaces in the large N limit

TL;DR

This paper studies Gaussian distributions on Riemannian symmetric spaces, focusing on approximating partition functions in high-dimensional limits, with applications to SPD matrices and the Siegel domain, inspired by concepts from physics.

Contribution

It introduces large N asymptotic approximations for partition functions of Gaussian distributions on symmetric spaces, providing explicit formulas and saddle-point characterizations.

Findings

Exact partition function formulas for certain spaces

Asymptotic approximations improve with increasing dimension

Saddle-point equations characterize the large N limit of the Siegel domain

Abstract

We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some cases, such as the space of Hermitian positive definite matrices or hyperbolic space, it is possible to compute them exactly using techniques from random matrix theory. However, in most cases which are important to applications, such as the space of symmetric positive definite (SPD) matrices or the Siegel domain, this is only possible numerically. Moreover, when we consider, for instance, high-dimensional SPD matrices, the known algorithms for computing partition functions can become exceedingly slow. Motivated by notions from theoretical physics, we will discuss how to approximate the partition functions in the large $N$ limit: an approximation that gets increasingly better as the dimension of the underlying symmetric space (more precisely, its rank) gets larger. We will give formulas for leading order terms in the case of SPD matrices and related spaces. Furthermore, we will characterize the large $N$ limit of the Siegel domain through a singular integral equation arising as a saddle-point equation.