Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-30T00:16:11.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-intersections of random walks on discrete groups

Published online by Cambridge University Press:  24 October 2008

David J. Aldous
David J. Aldous
Affiliation:
Department of Statistics, University of California, Berkeley, CA 94720, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

For a random walk on a finite group, the distribution after n steps will converge, as n→∞, to the uniform distribution (under mild conditions). The asymptotic behaviour of such walks can be studied easily using standard Markov chain theory. But there are many natural problems about the non-asymptotic behaviour which have no simple solution in general. For example, what is

the time until a specified element is first visited?

the time until all element have been visited?

the time until the distribution approaches the uniform distribution?

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 155 - 177
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Aldous, D. J.. Some inequalities for reversible Markov chains. J. Lond. Math. Soc. 25 (1982), 564–576.CrossRefGoogle Scholar
[2]Aldous, D. J.. On the time taken by random walks on finite groups to visit every state. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 62 (1983), 361–374.CrossRefGoogle Scholar
[3]Aldous, D. J.. Random walks on finite groups and rapidly mixing Markov chains. In Séminaire de Probabilites XVII, (Lecture Notes in Mathematics 986). (Springer, 1983), 243–297.CrossRefGoogle Scholar
[4]Aldous, D. J.. Minimization algorithms and random walk on the d-cube. Ann. Probab. 11 (1983), 403–413.CrossRefGoogle Scholar
[5]Aldous, D. J.. Self-intersections of a one-dimensional random walk. (Preprint, 1984.)Google Scholar
[6]Diaconis, P.. Group Theory in Statistics. I.M.S. (Lecture Notes in Statistics) (to appear, 1984).Google Scholar
[7]Diaconis, P. and Shahshahani, M.. Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 57 (1981), 159–179.CrossRefGoogle Scholar
[8]Flatto, L., Odlyzko, A. M. and Wales, D. B.. Random shuffles and group representations. Ann. Probab. 13 (1985), 154–178.CrossRefGoogle Scholar
[9]Kendall, D. G.. Unitary dilations of Markov transition operators, and the corresponding integral representations for transition-probability matrices. In Probability and Statistics (The Harold Cramer Volume) (Wiley, 1960).Google Scholar
[10]Letac, G.. Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov. Adv. in Appl. Probab. 14 (1982), 272–294.CrossRefGoogle Scholar
[11]Letac, G. and Takacs, L.. Random walks on the m-dimensional cube. J. Reine Angew. Math. 310 (1979), 178–185.Google Scholar
[12]Silverman, B. W. and Brown, T. C.. Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15 (1978), 815–825.CrossRefGoogle Scholar
[13]Zubhov, A. M. and Mikhailov, V. G.. Limit distribution of random variables associated with long duplications in a sequence of independent trials. Theory Probab. Appl. 19 (1974), 172–179.Google Scholar