p -adic boundary laws and Markov chains on trees
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p-adic boundary laws and Markov chains on trees A. Le Ny1 · L. Liao1 · U. A. Rozikov2 Received: 10 July 2019 / Revised: 27 June 2020 / Accepted: 16 July 2020 © Springer Nature B.V. 2020
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. Keywords Cayley trees · Boundary laws · Gibbs measures · Translation invariant measures · p-adic numbers · p-adic probability measures · p-adic Markov chain · Non-Archimedean probability Mathematics Subject Classification 46S10 · 82B26 · 12J12 (primary); 60K35 (secondary)
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U. A. Rozikov [email protected] A. Le Ny [email protected] L. Liao [email protected]
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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
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Institute of Mathematics, 81, Mirzo Ulug’bek Str., 100170 Tashkent, Uzbekistan
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A. Le Ny et al.
1 Introduction In this paper, we develop a boundary law argument to study p-adic Markov chains on general trees. In the real case Markov chains on trees are particular cases of Gibbs measures corresponding to a Hamiltonian with nearest-neighbor interactions. In the theory of Gibbs measures on trees (see [9, Chapter 12] and [23]) the main problem is to describe the set of limiting Gibbs measures corresponding to a given Hamiltonian. A complete analysis of this set is often a difficult problem, this is even not completely described for the Ising model (see [4–6,25] for some recent results). Parallel to the real-valued Gibbs measures, the p-adic Gibbs measures are studied using the p-adic mathematical physics in [3,13,14,24,30]. A p-adic distribution is an analogue of ordinary distributions that takes values in a ring of p-adic numbers [1,12,13]. Analogically to a measure on a measurable space, a p-adic measure is a special case of a p-adic distribution. A p-adic distribution taking values in a normed space is called a p-adic measure if the values on compact open subsets are bounded. It is known that some p-adic models in physics cannot be described using ordinary Kolmogorov’s probability theory [14,16,18,30]. In [15] the p-adic probability theory was developed using the theory of non-Archimedean measures [29]. In [7,11,19– 21,26] various models of statistical physics in the conte
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