Random Walks in Compact Groups

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.

2010 Mathematics Subject Classification: 60B15, 22E30, 05E15

Keywords and Phrases: random walk, spectral gap, diameter, poly-log, Solovay-Kitaev, compact group, Cayley graph

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