The Asymptotic Statistics of Random Covering Surfaces

被引:3
|
作者
Magee, Michael [1 ]
Puder, Doron [2 ]
机构
[1] Univ Durham, Dept Math Sci, Durham, England
[2] Tel Aviv Univ, Sch Math Sci, Tel Aviv, Israel
来源
FORUM OF MATHEMATICS PI | 2023年 / 11卷
基金
以色列科学基金会; 欧洲研究理事会;
关键词
20C15; 20B30; 20P05; 20F34; 20F65; 20F70; 60B15; SUBGROUP GROWTH; MODULI SPACES; RIEMANN; CURVES; NUMBER;
D O I
10.1017/fmp.2023.13
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Let Gg 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(Gg, 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 f E Xg,n and y E Gg, we let fixy(f) be the number of fixed points of the permutation f(y). The function fixy 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 ? co, for the expectation of fixy with respect to the uniform probability measure on Xg,n, which is denoted by Eg,n [fixy]. We prove that if y E Gg is not the identity and q is maximal such that y is a qth power in Gg, then Eg,n [fixy] = d(q) + O(n-1) as n ? co, where d (q) is the number of divisors of q. Even the weaker corollary that Eg,n[fixy] = o(n) as n ? co is a new result of this paper. We also prove that Eg,n [fixy] can be approximated to any order O(n-M) by a polynomial in n-1.
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页数:51
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