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p-adic boundary laws and Markov chains on trees

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Abstract

In this paper, we consider a potential on general infinite trees with q spin values and nearest-neighbor p-adic interactions given by a stochastic matrix. We show the uniqueness of the associated Markov chain (splitting Gibbs measures) under some sufficient conditions on the stochastic matrix. Moreover, we find a family of stochastic matrices for which there are at least two p-adic Markov chains on an infinite tree (in particular, on a Cayley tree). When the p-adic norm of q is greater (resp. less) than the norm of any element of the stochastic matrix then it is proved that the p-adic Markov chain is bounded (resp. is not bounded). Our method uses a classical boundary law argument carefully adapted from the real case to the p-adic case, by a systematic use of some nice peculiarities of the ultrametric (p-adic) norms.

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Notes

  1. Compare with real boundary law of [9, Definition (12.10)].

  2. http://www.numbertheory.org/php/p-adic.html.

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Acknowledgements

UAR thanks the University Paris-Est Créteil (UPEC) for the hospitality during June 2019, where this work has been achieved, and Labex Bézout (Université Paris Est) for the financial and logistic support of this visit. The collaboration of the authors is realized within the project “Real/ p-adic dynamical systems and Gibbs measures” funded by LabEx Bézout (ANR-10-LABX-58). We thank referee for helpful comments.

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  1. Laboratoire d’Analyse et de Mathématiques Appliquées, LAMA UMR CNRS 8050, UPEC, Université Paris-Est, 61 Avenue du Général de Gaulle, 94010, Créteil cedex, France

    A. Le Ny & L. Liao

  2. Institute of Mathematics, 81, Mirzo Ulug’bek Str., 100170, Tashkent, Uzbekistan

    U. A. Rozikov

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  1. A. Le Ny
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  2. L. Liao
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  3. U. A. Rozikov
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Correspondence to U. A. Rozikov.

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Ny, A.L., Liao, L. & Rozikov, U.A. p-adic boundary laws and Markov chains on trees. Lett Math Phys 110, 2725–2741 (2020). https://doi.org/10.1007/s11005-020-01316-7

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