The Asymptotic Statistics of Random Covering Surfaces

Abstract

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