Random Unitary Representations of Surface Groups I: Asymptotic Expansions

Michael Magee

doi:10.1007/s00220-021-04295-5

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 gge 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.

Highlights

Let g ∈ N with g ≥ 2 and let g denote a closed topological surface of genus g
The chiral expansion means that the (ρ, W ) are parameterized by where λ runs over Young diagrams
Young diagrams A Young diagram (YD) is a collection of left-aligned rows of identical square boxes in the plane, where the number of boxes in each row is non-increasing from top to bottom

Summary

Introduction

Let g ∈ N with g ≥ 2 and let g denote a closed topological surface of genus g. It is quite well understood that matrix integrals such as I(w, ρ) are challenging in this regime as the main method of performing such integrals, known as the Weingarten calculus, often fails to produce understandable answers there This is because the Weingarten function Wgn,k defined in (2.12) becomes increasingly complicated for k n, drawing on more and more different representations of large symmetric groups. The main idea is that after some splitting up, parts of I(w, ρ) can be evaluated by integrating first tr(ρ(Rg(x))) over all double cosets for a very large subgroup U(n − D) ≤ U(n) where D is bounded depending only on w, which is fixed During this first integration, the structure of the word Rg can be exploited to produce a lot of cancelation. (dim W )I(w, ρ) + OB,w,g n|w|n−2 log B

Related works I

Related works II

Background

Organization of Representations

The Contribution from a Single Family of Representations

The Total Contribution from Large-Dimensional Families