The Asymptotic Statistics of Random Covering Surfaces

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.