The Asymptotic Statistics of Random Covering Surfaces

Let Γ g 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(Γ g,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 ∈ Γ g , 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 ∈ Γ g is not the identity and q is maximal such that γ is a q th power in Γ g , then align*Eg,n[fixγ]=d(q)+O(n⁻¹) align* 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 .