Random Unitary Representations of Surface Groups I: Asymptotic expansions

In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let ${\mathrm{\Sigma }}_g$ denote a topological surface of genus $g\ge 2$. We establish the existence of a large $n$ asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of ${\pi }_1({\mathrm{\Sigma }}_g)$ under a random representation of ${\pi }_1({\mathrm{\Sigma }}_g)$ into $\mathsf{S}\mathsf{U}(n)$. Each such expected value involves a contribution from all irreducible representations of $\mathsf{S}\mathsf{U}(n)$. The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.