Electronic Journal of Probability

Random Walks On Finite Groups With Few Random Generators

Igor Pak

Full-text: Open access

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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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