Professor Michael Magee michael.r.magee@durham.ac.uk
Professor
The Asymptotic Statistics of Random Covering Surfaces
Magee, Michael; Puder, Doron
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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Copyright Statement
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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