The Asymptotic Statistics of Random Covering Surfaces

Abstract Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface genus $g\geq 2$ . We develop new method for integrating over representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is symmetric permutations $\{1,\ldots ,n\}$ Equivalently, this all vertex-labeled, n -sheeted covering spaces g Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \Gamma we let $\mathsf {fix}_{\gamma }(\phi )$ number fixed points permutation (\gamma The function }$ special case natural family functions on called Wilson loops. Our methodology leads to an asymptotic formula, as $n\to \infty $ expectation with respect uniform probability measure which denoted by {E}_{g,n}[\mathsf }]$ prove that if not identity q maximal such th power in then $$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$ $d\left (q\right divisors Even weaker corollary }]=o(n)$ result paper. also can approximated any order $O(n^{-M})$ polynomial $n^{-1}$