Random Walks with Few Random Generators

Let G be a nite 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 jGj, a > 1, this walk mixes in under m = 2a log a a?1 log jGj steps. A similar result was obtained earlier by Alon and Roichman (see AR]), Dou and Hildebrand (see DH]) using a diierent techniques. We also prove that when sets are of size k = log 2 jGj+O(loglogjGj), m = O(log 3 jGj) steps suuce for mixing of the corresponding symmetric lazy random walk. Finally, when G is abelian we obtain better bounds in both cases.