Word Measures on Unitary Groups

Abstract

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free group $F_r$ on $r$ generators determines a word map from $U(n{)}^r$ to $U(n)$, defined by substitutions. The $w$-measure on $U(n)$ is defined as the pushforward via this word map of the Haar measure on $U(n{)}^r$. Let $T{r}_w(n)$ denote the expected trace of a random unitary matrix sampled from $U(n)$ according to the $w$-measure. It was shown by Voiculescu [Voic 91'] that for $w\ne 1$ this expected trace is $o(n)$ asymptotically in $n$. We relate the numbers $T{r}_w(n)$ to the theory of commutator length of words and obtain a much stronger statement: $T{r}_w(n)=O(n^{1-2g})$, where $g$ is the commutator length of $w$. Moreover, we analyze the number ${\lim }_{n\to \infty }{n}^{2g-1}\cdot T{r}_w(n)$ and show it is an integer which, roughly, counts the number of (equivalence classes of) solutions to the equation $[u_1,v_1]...[u_g,v_g]=w$ with $u_i,v_i\in F_r$. Similar results are obtained for finite sets of words and their commutator length, and we deduce that one can 'hear' the stable commutator length of a word by 'listening' to its unitary measures.

68 pages, 13 figures, results much more general than previous version (see Section 1.2)

References in corpus (4)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Expansion of Random Graphs: New Proofs, New Results
• Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants
• Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries

Cited by in corpus (4)

• 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 $\mathrm{Sp}\left(n\right)$
• Random Unitary Representations of Surface Groups II: The large $n$ limit
• Commutators of random matrices from the unitary and orthogonal groups