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Some Properties of Markov chains on the Free Group $${\mathbb {F}}_2$$ F 2

Author

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  • Antoine Goldsborough

    (Heriot-Watt University)

  • Stefanie Zbinden

    (Heriot-Watt University)

Abstract

Random walks cannot, in general, be pushed forward by quasi-isometries. Tame Markov chains were introduced as a ‘quasi-isometry invariant’ generalization of random walks. In this paper, we construct several examples of tame Markov chains on the free group exhibiting ‘exotic’ behavior; one, where the drift is not well defined and one where the drift is well defined but the Central Limit Theorem does not hold. We show that this is not a failure of the notion of tame Markov chain, but rather that any quasi-isometry invariant theory that generalizes random walks will include examples without well-defined drift.

Suggested Citation

  • Antoine Goldsborough & Stefanie Zbinden, 2024. "Some Properties of Markov chains on the Free Group $${\mathbb {F}}_2$$ F 2," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2330-2351, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-023-01294-1
    DOI: 10.1007/s10959-023-01294-1
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