Abstract
We study the rate of convergence of symmetric random walks on finite groups to the uniform distribution. A notion of moderate growth is introduced that combines with eigenvalue techniques to give sharp results. Roughly, for finite groups of moderate growth, a random walk supported on a set of generators such that the diameter of the group is γ requires order γ2 steps to get close to the uniform distribution. This result holds for nilpotent groups with constants depending only on the number of generators and the class. Using Gromov's theorem we show that groups with polynomial growth have moderate growth.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AD]D. Aldous, P. Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93 (1986), 333–348.
[B]H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math Soc. 25 (1982), 603–614.
[C]T. Carne, A transmutation formula for Markov chains, Bull. Sc. Math. 2nd série 109 (1985), 399–405.
[ChDG]F. Chung, P. Diaconis, R. L. Graham, A random walk problem arising in random number generation, Ann. Prob. 15 (1987), 1148–1165.
[CoS-C]T. Coulhon, L. Saloff-Coste, Puissance d'un operateur regularisant, Ann. Inst. H. Poincare Proba. Stat. 26 (1990), 419–436.
[D]P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA (1988).
[DG]P. Diaconis, R. L. Graham, An affine walk on the hypercube, Jour. Comp. Appl. Math 41 (1992), 215–235.
[DS-C1]P. Diaconis, L. Saloff-Coste, Nash's inequality and convergence of finite Markov chains to equilibrium, Technical report (1992).
[DS-C2]P. Diaconis, L. Saloff-Coste, Comparison theorems for random walk on groups, To appear in Ann. Prob. 21 (1993).
[DS-C3]P. Diaconis, L. Saloff-Coste, An application of Harnack inequalities to random walk on nilpotent quotients, Technical report, Dept. of Mathematics, Harvard University (1993).
[DStr]P. Diaconis, D. Stroock, Geometric bounds for reversible Markov chains, Ann. Appl. Prob. 1 (1991), 36–61.
[E]J. Ellenberg, A sharp diameter bound for upper triangular matrices, Senior honors thesis, Dept. Math., Harvard University (1993).
[Gr]M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53 (1981), 53–73.
[Gu]Y. Guivarc'h, Croissance polynomiale et periodes des fonctions harmoniques, Bull. Soc. Math. France, 101 (1973), 333–379.
[H]M. Hall, The Theory of Groups, 2nd ed, Chelsea, New York (1976).
[Ha]P. Hall, Nilpotent Groups. In Collected Works of Philip Hall, Oxford University Press, Oxford (1957), 417–462.
[He]W. Hebisch, On heat kernels on Lie groups.Math Zeit. 210 (1992), 593–605.
[HeS-C]W. Hebisch, L. Saloff-Coste, Gaussian estimates for Markov chains, Ann. Proba. 21 (1993), 673–709.
[Hi]M. Hildebrand, Random processes of the formX n+1=a n X n+b n (modp), Ann. Proba. 21 (1993), 710–720.
[IR]K. Ireland, M. Rosen, Elements of Number Theory, Bogdon and Quigley, Tarrytown, New York (1972).
[J]D.L. Johnson, The groups of formal power series under substitutions, Jour. Austral. Math Soc. (A) 45 (1988), 296–302.
[MKSo]W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, 2nd ed., Dover, New York (1972).
[Ro]J. Rotman, The Theory of Groups: An Introduction, 3rd ed., Allyn and Bacon, Boston (1984).
[St]R. Stong, A random walk on the Heisenberg group, Technical report, Department of Mathematics, U.C.L.A. (1992).
[Su1]M. Suzuki, Group Theory I, Springer Verlag, New York (1982).
[Su2]M. Suzuki, Group Theory II, Springer Verlag, New York (1986).
[T]J. Tits, Appendix to Gromov's paper, Publ. Math. I.H.E.S. 58 (1981), 74–78.
[V]N. Varopoulos, Long range estimates for Markov chains, Bull. Sc. Math. 109 (1985), 225–252.
[VS-CCo]N. Varopoulos, L. Saloff-Coste, Th. Coulhon, Analysis and geometry on groups, Cambridge University Press, Cambridge (1993).
[Z]M. Zack, A random walk on the Heisenberg group. Ph.D Thesis, Dept. Math., University of California, La Jolla (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Diaconis, P., Saloff-Coste, L. Moderate growth and random walk on finite groups. Geometric and Functional Analysis 4, 1–36 (1994). https://doi.org/10.1007/BF01898359
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01898359