Local Statistics of Random Permutations from Free Products

Puder, Doron; Zimhoni, Tomer

doi:10.48550/arxiv.2203.12250

Cited by 1 publication

(1 citation statement)

References 27 publications

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“…In this § §2.1, we summarize some definitions and results from [MP22a].8 7Example 3.16 and Figure 3.1 in [PZ22] illustrate that the coefficient of 𝑛 𝜒 (𝑌 ) in E emb 𝑛 (𝑌 ) for Y boundary reduced may indeed be strictly larger than 1. In fact, [PZ22] proves it is always a positive integer.…”

Section: Tiled Surfaces and Core Surfacesmentioning

confidence: 98%

The Asymptotic Statistics of Random Covering Surfaces

Magee

Puder

2023

Forum of Mathematics, Pi

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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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Section: Tiled Surfaces and Core Surfacesmentioning

confidence: 98%