  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 by their 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.
