# $p$-adic boundary laws and Markov chains on trees

## TL;DR

This paper investigates $p$-adic boundary laws and Markov chains on infinite trees, establishing conditions for uniqueness, existence of multiple chains, and boundedness properties using adapted boundary law methods in the $p$-adic setting.

## Contribution

It introduces a systematic adaptation of boundary law techniques to the $p$-adic context, revealing new phenomena in Markov chains on trees.

## Key findings

- Uniqueness of $p$-adic Markov chains under certain conditions.
- Existence of multiple $p$-adic Markov chains for specific stochastic matrices.
- Boundedness of chains depending on the $p$-adic norm relations.

## 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.