The Non-Commutative Cycle Lemma

TL;DR

This paper introduces a non-commutative cycle lemma applicable to free groups, providing new solutions to cyclic reduction problems and insights into random matrix fluctuations, especially Kesten's Law.

Contribution

It develops a non-commutative version of the cycle lemma and applies it to free groups and random matrix theory, extending classical combinatorial results.

Findings

Non-commutative cycle lemma for free groups

Solutions to cyclic reduction problems in free groups

Analysis of fluctuations in Kesten's Law

Abstract

We present a non-commutative version of the cycle lemma of Dvoretsky and Motzkin that applies to free groups and use this result to solve a number of problems involving cyclic reduction in the free group. We also describe an application to random matrices, in particular the fluctuations of Kesten's Law.