The Asymptotic Statistics of Random Covering Surfaces

Magee, Michael; Puder, Doron

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