Let $H$ be a hermitian $N\times N$ random matrix from the gaussian unitary ensemble. Then it is a standard result that, in the large-$N$ limit, $${\mathbb E}\left[\frac1N{\rm Tr}\,H^{2k}\right]=c_k,$$ where $c_k$ is the the $k$th Catalan number.
Question: is there a formula for the variance of $\frac1N {\rm Tr}\,H^{2k}$?
It seems like such a formula should be derivable from the eigenvalue correlation formula, but I cannot find a closed expression in the literature.
