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The Asymptotic Statistics of Random Covering Surfaces

Magee, Michael; Puder, Doron

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Authors

Doron Puder



Abstract

Let Γg be the fundamental group of a closed connected orientable surface of genus g ≥ 2. We develop a new method for integrating over the representation space Xg,n = Hom(Γg, Sn) where Sn is the symmetric group of permutations of {1, . . . , n}. Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given ϕ ∈ Xg,n and γ ∈ Γg, we let fixγ(ϕ) be the number of fixed points of the permutation ϕ(γ). The function fixγ is a special case of a natural family of functions on Xg,n called Wilson loops. Our new methodology leads to an asymptotic formula, as n → ∞, for the expectation of fixγ with respect to the uniform probability measure on Xg,n, which is denoted by Eg,n[fixγ]. We prove that if γ ∈ Γg is not the identity, and q is maximal such that γ is a qth power in Γg, then Eg,n [fixγ] = d(q) + O(n−1) as n → ∞, where d (q) is the number of divisors of q. Even the weaker corollary that Eg,n[fixγ] = o(n) as n → ∞ is a new result of this paper. We also prove that Eg,n[fixγ] can be approximated to any order O(n−M) by a polynomial in n−1.

Citation

Magee, M., & Puder, D. (2023). The Asymptotic Statistics of Random Covering Surfaces. Forum of mathematics. Pi, 11, Article e15. https://doi.org/10.1017/fmp.2023.13

Journal Article Type Article
Acceptance Date Mar 27, 2023
Online Publication Date May 15, 2023
Publication Date 2023
Deposit Date Apr 20, 2020
Publicly Available Date Jun 23, 2023
Journal Forum of Mathematics, Pi
Peer Reviewed Peer Reviewed
Volume 11
Article Number e15
DOI https://doi.org/10.1017/fmp.2023.13

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