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.
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Sinai, Y.G. Simple Random Walks on Tori. Journal of Statistical Physics 94, 695–708 (1999). https://doi.org/10.1023/A:1004564824697
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DOI: https://doi.org/10.1023/A:1004564824697