The Asymptotic Statistics of Random Covering Surfaces

Let be the fundamental group of a closed connected orientable surface of genus. We develop a new method for integrating over the representation space, where is the symmetric group of permutations of. Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given and, we let be the number of fixed points of the permutation. The function is a special case of a natural family of functions on called Wilson loops. Our new methodology leads to an asymptotic formula, as, for the expectation of with respect to the uniform probability measure on, which is denoted by. We prove that if is not the identity and q is maximal such that is a q th power in, then as, where is the number of divisors of q. Even the weaker corollary that as is a new result of this paper. We also prove that can be approximated to any order by a polynomial in.