Open Access
1999 Random Walks On Finite Groups With Few Random Generators
Igor Pak
Author Affiliations +
Electron. J. Probab. 4: 1-11 (1999). DOI: 10.1214/EJP.v4-38


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.


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Igor Pak. "Random Walks On Finite Groups With Few Random Generators." Electron. J. Probab. 4 1 - 11, 1999.


Accepted: 11 November 1998; Published: 1999
First available in Project Euclid: 4 March 2016

zbMATH: 0918.60061
MathSciNet: MR1663526
Digital Object Identifier: 10.1214/EJP.v4-38

Primary: 60C05
Secondary: 60J15

Keywords: probabilistic method , Random random walks on groups , random subproducts , separation distance

Vol.4 • 1999
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