### Abstract

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 x - ω 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.

Original language | English (US) |
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Pages (from-to) | 695-708 |

Number of pages | 14 |

Journal | Journal of Statistical Physics |

Volume | 94 |

Issue number | 3-4 |

DOIs | |

State | Published - Feb 1999 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Homological equation
- Levy excursion
- Markov chain
- Stable law

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## Cite this

Sinai, Y. G. (1999). Simple random walks on tori.

*Journal of Statistical Physics*,*94*(3-4), 695-708. https://doi.org/10.1023/a:1004564824697