The Asymptotic Statistics of Random Covering Surfaces

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