Commutators of random matrices from the unitary and orthogonal groups

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