p-Adic Markov Chains with Countable State on Trees

In this paper, we investigate spin systems on general infinite trees. The spins can take countably many values, and nearest-neighbor interactions are governed by a p -adic stochastic matrix. We establish sufficient conditions on the stochastic matrix that guarantee the uniqueness of the associated Markov chain. Furthermore, we identify a family of stochastic matrices that lead to the existence of at least two distinct p -adic Markov chains on an infinite tree, particularly a Cayley tree.