  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 $g\geq2$. 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 $\mathsf{SU}(n)$. Each
such expected value involves a contribution from all irreducible
representations of $\mathsf{SU}(n)$. 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.
