Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$

TL;DR

This paper explores the relationship between random unitary matrices, surface topology, and mapping class groups, revealing new connections and asymptotic behaviors in matrix integrals related to free group words.

Contribution

It introduces a novel framework linking moments of measures on unitary groups to surfaces and mapping class groups, extending the understanding of matrix integrals and free group equations.

Findings

Derived asymptotic bounds on moments of unitary measures

Connected solutions of free group equations to surface topology

Showed how to determine stable commutator length from unitary measures

Abstract

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n\right)$ of unitary matrices. More concretely, we study measures induced by free words on $\mathcal{U}\left(n\right)$. Let $F_{r}$ be the free group on $r$ generators. To sample a random element from $\mathcal{U}\left(n\right)$ according to the measure induced by $w\in F_{r}$, one substitutes the $r$ letters in $w$ by $r$ independent, Haar-random elements from $\mathcal{U}\left(n\right)$. The main theme of this paper is that every moment of this measure is determined by families of pairs $\left(\Sigma,f\right)$, where $\Sigma$ is an orientable surface with boundary, and $f$ is a map from $\Sigma$ to the bouquet of $r$ circles, which sends the boundary components of $\Sigma$ to powers of $w$. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of $\Sigma$. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on $\mathcal{U}\left(n\right)$ bears information about the number of solutions to the equation $\left[u_{1},v_{1}\right]\cdots\left[u_{g},v_{g}\right]=w$ in the free group, and deduce that one can ``hear'' the stable commutator length of a word through its unitary word measures.