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.

中文翻译：

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树上的 p-adic 边界定律和马尔可夫链

在本文中，我们考虑具有由随机矩阵给出的最近邻 $p$-adic 相互作用的一般无限树上的 $q$-状态势。{我们在随机矩阵的一些充分条件下展示了相关马尔可夫链的唯一性（{\em 分裂吉布斯测度}）。此外，我们找到了一个随机矩阵族，其中至少有两个 $p$-adic无限树上的马尔可夫链（特别是在凯莱树上）。当 $q$ 的 $p$-adic 范数大于（{\em resp.} 小于）随机矩阵的任何元素的范数时，证明 $p$-adic 马尔可夫链是有界的（{ \em resp.} 是无界的）。我们的方法{使用}一个经典的边界律论证，通过系统地使用超度量 ($p$-adic) 规范的一些很好的特性，从实际情况精心改编为 $p$-adic 情况。