Random walks in compact groups

TL;DR

This paper investigates how quickly random walks on compact groups converge to the uniform distribution, providing improved estimates for finite and semi-simple Lie groups, advancing understanding of their mixing times.

Contribution

It offers new poly-logarithmic bounds on convergence speed for random walks on finite and semi-simple Lie groups, improving previous results by several researchers.

Findings

Poly-logarithmic convergence estimates for finite groups.

Enhanced bounds for compact semi-simple Lie groups.

Improved understanding of mixing times in compact groups.

Abstract

Let X_1,X_2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.