Let X be an orientable hyperbolic surface of genus \(g\ge 2\) with a marked point o, and let \(\Gamma \) be an orientable hyperbolic surface group isomorphic to \(\pi _{1}(X,o)\) . Consider the space \(\text {Hom}(\Gamma ,S_{n})\) which corresponds to n-sheeted covers of X with labeled fiber. Given \(\gamma \in \Gamma \) and a uniformly random \(\phi \in \text {Hom}(\Gamma ,S_{n})\) , what is the expected number of fixed points of \(\phi (\gamma )\) ? Formally, let \(F_{n}(\gamma )\) denote the number of fixed points of \(\phi (\gamma )\) for a uniformly random \(\phi \in \text {Hom}(\Gamma ,S_{n})\) . We think of \(F_{n}(\gamma )\) as a random variable on the space \(\text {Hom}(\Gamma ,S_{n})\) . We show that an arbitrary fixed number of products of the variables \(F_{n}(\gamma )\) are asymptotically independent as \(n\rightarrow \infty \) when there are no obvious obstructions. We also determine the limiting distribution of such products. Additionally, we examine short cycle statistics in random permutations of the form \(\phi (\gamma )\) for a uniformly random \(\phi \in \text {Hom}(\Gamma ,S_{n})\) . We show a similar asymptotic independence result and determine the limiting distribution.
