Klara Courteaut: Multivariate normal approximation of traces of powers of random orthogonal and symplectic matrices

Abstract

In '94 Diaconis and Shahshahani computed the joint moments of the traces of powers of orthogonal, unitary and symplectic matrices, distributed according to Haar measure. They found that they are equal up to high order to the joint moments of independent (complex) Gaussians. This shows that the traces converge to independent Gaussians as the size of the matrix goes to infinity, and also suggests that the rate should be rather fast. I will discuss a recent result on this rate of convergence obtained with Kurt Johansson. We give a super-exponential bound on the total variation distance between the traces of the first \(m\) powers of an \(n \times n\) orthogonal/symplectic matrix and an \(m\)-dimensional Gaussian vector, where \(m\) is allowed to grow and where the constants are explicit.