Journal of Statistical Physics

, Volume 94, Issue 3–4, pp 695–708 | Cite as

Simple Random Walks on Tori

  • Ya. G. Sinai


We consider a Markov chain whose phase space is a d-dimensional torus. A point x jumps to x+ω with probability p(x) and to xω with probability 1−p(x). For Diophantine ω and smooth p we prove that this Markov chain has an absolutely continuous invariant measure and the distribution of any point after n steps converges to this measure.

Markov chain homological equation Levy excursion stable law 


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  1. [F]
    Feller, W., Introduction to Probability Theory and Its Applications, Vol. 1 & 2 (Wiley, New York, 1971).Google Scholar
  2. [GK]
    Gnedenko, B. V., and Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, MA, 1968).Google Scholar
  3. [S1]
    Sinai, Ya. G., The limiting behavior of one-dimensional random walks in random media, Probability Theory and Its Applications 27:247–258 (1982).Google Scholar
  4. [S2]
    Sinai, Ya. G., Distribution of some functionals of the integral of the brownian motion, Theor. and Math. Physics (in Russian) 90:323–353 (1992).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Ya. G. Sinai
    • 1
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrinceton
  2. 2.Landau Institute of Theoretical PhysicsMoscowRussia

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