  Gaussian distributions can be generalized from Euclidean space to a wide
class of Riemannian manifolds. Gaussian distributions on manifolds are harder
to make use of in applications since the normalisation factors, which we will
refer to as partition functions, are complicated, intractable integrals in
general that depend in a highly non-linear way on the mean of the given
distribution. Nonetheless, on Riemannian symmetric spaces, the partition
functions are independent of the mean and reduce to integrals over finite
dimensional vector spaces. These are generally still hard to compute
numerically when the dimension (more precisely the rank $N$) of the underlying
symmetric space gets large. On the space of positive definite Hermitian
matrices, it is possible to compute these integrals exactly using methods from
random matrix theory and the so-called Stieltjes-Wigert polynomials. In other
cases of interest to applications, such as the space of symmetric positive
definite (SPD) matrices or the Siegel domain (related to block-Toeplitz
covariance matrices), these methods seem not to work quite as well.
Nonetheless, it remains possible to compute leading order terms in a large $N$
limit, which provide increasingly accurate approximations as $N$ grows. This
limit is inspired by realizing a given partition function as the partition
function of a zero-dimensional quantum field theory or even Chern-Simons
theory. From this point of view the large $N$ limit arises naturally and
saddle-point methods, Feynman diagrams, and certain universalities that relate
different spaces emerge.
