Geodesic
The Asymptotic Statistics of Random Covering Surfaces
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

Let $\Gamma _{g}$ 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 genus g. 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 and q is maximal such that $\gamma $ is a qth power 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 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 $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$. 
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     author = {Michael Magee and Doron Puder},
     title = {The {Asymptotic} {Statistics} of {Random} {Covering} {Surfaces}},
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Michael Magee; Doron Puder. The Asymptotic Statistics of Random Covering Surfaces. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.13

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