The Asymptotic Statistics of Random Covering Surfaces

Abstract

Let ${\Gamma }_g$ 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 ${\mathbb{X}}_{g,n}=\mathrm{Hom}({\Gamma }_g,S_n)$ 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 $\gamma \in {\Gamma }_g$, we let ${\mathsf{fix}}_{\gamma }(\varphi )$ be the number of fixed points of the permutation $\varphi (\gamma )$. 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, and $q$ is maximal such that $\gamma$ is a $q$th power in ${\Gamma }_g$, then \[ \mathbb{E}_{g,n}[\mathsf{fix}_γ]=d(q)+O(n^{-1}) \] as $n\to \infty$, where $d\left(q\right)$ is the number of divisors of $q$. 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 if $\gamma$ is not the identity then ${\mathbb{E}}_{g,n}[{\mathsf{fix}}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$.

51 pages, 6 figures. Slightly simplified Sections 5.2 and 5.4

References in corpus (6)

• Poisson--Dirichlet distribution for random Belyi surfaces
• A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$
• Word Measures on Symmetric Groups
• Random Unitary Representations of Surface Groups I: Asymptotic expansions
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Cited by in corpus (5)

• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• Extension of Alon's and Friedman's conjectures to Schottky surfaces
• Random Unitary Representations of Surface Groups II: The large $n$ limit
• Word Measures on $G{L}_N(q)$ and Free Group Algebras
• Degenerating hyperbolic surfaces and spectral gaps for large genus