Random Unitary Representations of Surface Groups I: Asymptotic expansions

TL;DR

This paper investigates the asymptotic behavior of random surface group representations into special unitary groups, providing detailed expansions and controlling contributions from various irreducible representations.

Contribution

It establishes large $n$ asymptotic expansions for expected traces of surface group elements under random $ ext{SU}(n)$ representations, with effective control over irreducible representation contributions.

Findings

Asymptotic expansions exist for expected trace values as n grows large.

Effective bounds are obtained for contributions from outside finite sets of representations.

The results apply to all fixed elements of the surface group.

Abstract

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.