Random walks in compact groups

Abstract

Let X1,X2, . . . be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X1 . . . X1 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.