Word Measures on Unitary Groups
Michael Magee, Doron Puder

TL;DR
This paper explores measures induced by free words on unitary groups, linking random matrix theory, free probability, and group theory to analyze traces and commutator lengths, revealing new algebraic and probabilistic connections.
Contribution
It establishes a quantitative relationship between expected traces of word-induced measures on unitary groups and the commutator length of words, providing new insights into their algebraic and probabilistic properties.
Findings
Expected trace $Tr_w(n)$ is $O(n^{1-2g})$ where $g$ is the commutator length.
The limit of $n^{2g-1} imes Tr_w(n)$ as $n o fty$ is an integer.
One can determine the stable commutator length of a word from its unitary measures.
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 . Every word in the free group on generators determines a word map from to , defined by substitutions. The -measure on is defined as the pushforward via this word map of the Haar measure on . Let denote the expected trace of a random unitary matrix sampled from according to the -measure. It was shown by Voiculescu [Voic 91'] that for this expected trace is asymptotically in . We relate the numbers to the theory of commutator length of words and obtain a much stronger statement: ,…
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Taxonomy
TopicsRandom Matrices and Applications · Advanced Combinatorial Mathematics · Advanced Algebra and Geometry
