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 Sigma(g) denote a topological surface of genus g >= 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) (Sigma(g)) under a random representation of pi(1 )(Sigma(g)) into SU(n). Each such expected value involves a contribution from all irreducible representations of SU(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.