Limit theorems for random matrix ensembles associated to symmetric spaces

This talk surveys recent work which develops aspects of classical random matrix theory in the broader framework of matrix ensembles associated to classical symmetric spaces. It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In joint work with Katrin Hofmann-Credner (Bochum), this has been generalized to random matrices taken from all infinitesimal versions of classical symmetric spaces. Like Wigner's, this result is universal in that it only depends on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. Joint work with Benoît Collins (Lyon/Ottawa) points into a different direction. Here random vectors of the form $(Tr(A^(1) V),..., Tr(A^(r) V))$ are studied, where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^(\nu)$ are deterministic parameter matrices. It is proven that for increasing matrix sizes these random vectors converge to a joint Gaussian limit. This generalizes work of Diaconis et al. on the compact classical groups.