Abstract

In this paper we investigate generalized Gibbs measure (GGM) for \(p\)-adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order \(k\geq 2\). This model is defined by \(p\)-adic parameters \(\lambda_i\), \(i\in \mathbb N\). We analyze \(p\)-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by \(p\)-adic version of Kolmogorov’s theorem. We define \(p\)-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our \(p\)-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters \(p\), \(k\) and \(\lambda_i\) we find the number of translation-invariant and two-periodic GGMs for the \(p\)-adic HC model on the Cayley tree of order two.

中文翻译：

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$$p$$ -Adic 算子的周期点及其 $$p$$ -Adic Gibbs 测度

摘要

在本文中，我们研究了\(p\) -adic 硬核 (HC) 模型的广义吉布斯测度 (GGM)，该模型在\(k\geq 2\)阶凯莱树上具有可数自旋值集。该模型由\(p\) -adic 参数\(\lambda_i\) , \(i\in \mathbb N\)定义。我们分析了\(p\) -adic 函数方程，它为有限维广义 Gibbs 分布提供了一致性条件。函数方程的每个解都通过Kolmogorov 定理的\(p\) -adic 版本定义了一个 GGM。我们定义\(p\)-adic Gibbs 分布作为有限维广义 Gibbs 分布一致族的极限，并表明，对于我们在 Cayley 树上的\(p\) -adic HC 模型，这样的 Gibbs 分布不存在。在参数\(p\)、\(k\)和\(\lambda_i\)的某些条件下，我们发现Cayley 上的\(p\) -adic HC 模型的平移不变和双周期 GGM 的数量二阶树。