Random Walks On Finite Groups With Few Random Generators

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 $k = 2a\, \log_2 |G|$, $a\gt1$, this walk mixes in under $m = 2a \,\log\frac{a}{a-1} \log |G|$ 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.