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Abstract <p> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic Markov chains on an infinite tree, particularly a Cayley tree. </p>

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\(p\)-Adic Markov Chains with Countable State on Trees

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • I. A. Sattarov

摘要

Abstract

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.