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 Electronic Journal of Probability > Vol. 4 (1999) > Paper 1 open journal systems 


Random Walks On Finite Groups With Few Random Generators

Igor Pak, Yale University


Abstract
Let $G$ be a finite group. Choose a set $S$ of size $k$ uniformly from $G$ and consider a lazy random walk on the corresponding Cayley graph. We show that for almost all choices of $S$ given $k = 2,a log_2 |G|$, $a>1$, this walk mixes in under $m = 2,a logfrac{a}{a-1} log |G|$ steps. A similar result was obtained earlier by Alon and Roichman and also by Dou and Hildebrand using a different techniques. We also prove that when sets are of size $k = log_2 |G| + O(log log |G|)$, $m = O(log^3 |G|)$ steps suffice for mixing of the corresponding symmetric lazy random walk. Finally, when $G$ is abelian we obtain better bounds in both cases.


Full text: PDF

Pages: 1-11

Published on: November 11, 1998
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Electronic Journal of Probability. ISSN: 1083-6489