p-adic boundary laws and Markov chains on trees

2019/07/05 by Ny, A. Le, Liao, L., Rozikov, U. A.
#12J12 #46S10 #60K35 #82B26 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1907.02854

Abstract

In this paper we consider q-state potential on general infinite trees with a nearest-neighbor p-adic interactions given by a stochastic matrix. We show the uniqueness of the associated Markov chain (\em 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 (\em resp. less) than the norm of any element of the stochastic matrix then it is proved that the p-adic Markov chain is bounded (\em 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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