On random matrices from the compact classical groups


If $M$ is a matrix taken randomly with respect to normalized Haar measure on $\mathrm{U}(n)$, $\mathrm{O}(n)$ or $\mathrm{Sp}(n)$, then the real and imaginary parts of the random variables $\mathrm{Tr}(M^k)$, $k\ge 1$, converge to independent normal random variables with mean zero and variance $k/2$, as the size $n$ o the matrix goes to infinity. For the unitary group this is a direct consequence of the strong Szegő theorem for Toeplitz determinants. We will prove a conjecture of Diaconis saying that for $\mathrm{U}(n)$ the rate of convergence to the limiting normal is $\mathrm{O}(n^{-\delta n})$ for some $\delta >0$, and for $\mathrm{O}(n)$ and $\mathrm{Sp}(n)$ it is $\mathrm{O}(e^{-cn})$ it is $\mathrm{O}(e^{-cn})$ for some $c>0$.


Kurt Johansson