Random Unitary Representations of Surface Groups I: Asymptotic expansions

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 $g\ge 2$. 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 ${\pi }_1({\Sigma }_g)$ 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.

45 pages

References in corpus (3)

• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• Random Unitary Representations of Surface Groups II: The large $n$ limit

Cited by in corpus (6)

• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• On the hyperbolic Bloch transform
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Random Unitary Representations of Surface Groups II: The large $n$ limit
• Moment Methods on compact groups: Weingarten calculus and its applications
• Word Measures on $G{L}_N(q)$ and Free Group Algebras