  Riemannian Gaussian distributions were initially introduced as basic building
blocks for learning models which aim to capture the intrinsic structure of
statistical populations of positive-definite matrices (here called covariance
matrices). While the potential applications of such models have attracted
significant attention, a major obstacle still stands in the way of these
applications: there seems to exist no practical method of computing the
normalising factors associated with Riemannian Gaussian distributions on spaces
of high-dimensional covariance matrices. The present paper shows that this
missing method comes from an unexpected new connection with random matrix
theory. Its main contribution is to prove that Riemannian Gaussian
distributions of real, complex, or quaternion covariance matrices are
equivalent to orthogonal, unitary, or symplectic log-normal matrix ensembles.
This equivalence yields a highly efficient approximation of the normalising
factors, in terms of a rather simple analytic expression. The error due to this
approximation decreases like the inverse square of dimension. Numerical
experiments are conducted which demonstrate how this new approximation can
unlock the difficulties which have impeded applications to real-world datasets
of high-dimensional covariance matrices. The paper then turns to Riemannian
Gaussian distributions of block-Toeplitz covariance matrices. These are
equivalent to yet another kind of random matrix ensembles, here called
"acosh-normal" ensembles. Orthogonal and unitary "acosh-normal" ensembles
correspond to the cases of block-Toeplitz with Toeplitz blocks, and
block-Toeplitz (with general blocks) covariance matrices, respectively.
