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

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.

arXiv comment53 pages, 12 figures, to appear in Inventiones Mathematicae

References in corpus (6)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Measure Preserving Words are Primitive
• Primitive Words, Free Factors and Measure Preservation
• Stable commutator length is rational in free groups
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

Cited by in corpus (13)

• Word Measures on Symmetric Groups
• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• The Asymptotic Statistics of Random Covering Surfaces
• Some Orbits of Free Words that are Determined by Measures on Finite Groups
• Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$
• Surface Words are Determined by Word Measures on Groups
• Random Unitary Representations of Surface Groups II: The large $n$ limit
• Commutators of random matrices from the unitary and orthogonal groups
• Word Measures on $G{L}_N(q)$ and Free Group Algebras
• Automorphism-invariant positive definite functions on free groups
• Moment Methods on compact groups: Weingarten calculus and its applications
• Wilson loop expectations as sums over surfaces on the plane
• On the Fourier coefficients of word maps on unitary groups