Electronic Journal of Probability

Random Walks On Finite Groups With Few Random Generators

Igor Pak

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60J15

Random random walks on groups random subproducts probabilistic method separation distance

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • D. Aldous and P. Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93, (1986), 333–348
  • D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. in Appl. Math. 8, (1987), no. 1, 69–97
  • D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph in preparation
  • N. Alon and Y. Roichman, Random Cayley graphs and expanders. Random Structures Algorithms 5, (1994), no. 2, 271–284
  • A. Astashkevich and I. Pak, Random walks on nilpotent and supersolvable groups, preprint, (1997)
  • L. Babai, Local expansion of vertex-transitive graphs and random geneartion in finite groups, Proc 23rd ACM STOC, (1991), 164–174
  • L. Babai, Automorphism groups, isomorphism, reconstruction. Handbook of combinatorics, Vol. 2, 1447–1540, Elsevier, Amsterdam, 1995.
  • L. Babai and G. Hetyei, On the diameter of random Cayley graphs of the symmetric group. Combin. Probab. Comput. 1, (1992), no. 3, 201–208.
  • P. Diaconis, Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes–-Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA, 1988.
  • P. Diaconis, The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 4, 1659–1664.
  • P. Diaconis, R. Graham and J. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms 1 (1990), no. 1, 51–72.
  • P. Diaconis, L. Saloff–Coste, Comparison techniques for random walk on finite groups. Ann. Probab. 21 (1993), no. 4, 2131–2156.
  • C. Dou and M. Hildebrand, Enumeration and random random walks on finite groups. Ann. Probab. 24, (1996), no. 2, 987–1000.
  • P. Erdős and R. R. Hall, Probabilistic methods in group theory. II. Houston J. Math. 2, (1976), no. 2, 173–180.
  • P. Erdős and A. Rényi, Probabilistic methods in group theory. J. Analyse Math. 14, (1965), 127–138.
  • W. Feller, An introduction to probability theory and its applications. Vol. I. Third edition John Wiley & Sons, Inc., New York, 1968.
  • A. Greenhalgh, A model for random random-walks on finite groups. Combin. Probab. Comput. 6 (1997), no. 1, 49–56.
  • M. Hildebrand, Random walks supported on random points of $\boldsymbol{Z}/n\boldsymbol{Z}$. Probab. Theory Related Fields 100 (1994), no. 2, 191–203.
  • W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group. Geom. Dedicata 36 (1990), no. 1, 67–87.
  • M. W. Liebeck and A. Shalev, The probability of generating a finite simple group. Geom. Dedicata 56 (1995), no. 1, 103–113.
  • I. Pak, Random walks on groups: strong uniform time approach, Ph.D. Thesis, Harvard U., (1997)
  • I. Pak, On finite geometric random walks, preprint, (1998)