Commutators of random matrices from the unitary and orthogonal groups

Abstract

We investigate the statistical properties of C = uvu−1 v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U( N) and O( N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the large- N limit, and the first N−1 correction is found.