## Random walks in compact groups

### Summary

Summary: Let $X_1,X_2,\ldots$ be independent identically distributed random elements of a compact group $G$. We discuss the speed of convergence of the law of the product $X_l\cdots X_1$ 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.

### Mathematics Subject Classification

60B15, 22E30, 05E15

### Keywords/Phrases

random walk, spectral gap, diameter, poly-log, Solovay-Kitaev, compact group, Cayley graph