## Electronic Journal of Probability

### Random Walks On Finite Groups With Few Random Generators

Igor Pak

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 4 (1999), paper no. 1, 11 pp.

Dates
Accepted: 11 November 1998
First available in Project Euclid: 4 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457125510

Digital Object Identifier
doi:10.1214/EJP.v4-38

Mathematical Reviews number (MathSciNet)
MR1663526

Zentralblatt MATH identifier
0918.60061

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60J15

Rights

#### Citation

Pak, Igor. Random Walks On Finite Groups With Few Random Generators. Electron. J. Probab. 4 (1999), paper no. 1, 11 pp. doi:10.1214/EJP.v4-38. https://projecteuclid.org/euclid.ejp/1457125510

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