Translation-Invariant Gibbs Measures for the Hard Core Model with a Countable Set of Spin Values

In this paper, we study the Hard Core (HC) model with a countable set $$\mathbb{Z}$$ of spin values on a Cayley tree of order $$k=2$$ . This model is defined by a countable set of parameters (that is, the activity function $$\lambda_{i}>0$$ , $$i\in\mathbb{Z}$$ ). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: Let $$k\geq 2$$ and $$\Lambda_{0}=\sum_{i}\lambda_{i}$$ . For $$\Lambda_{0}=+\infty$$ there is no translation-invariant Gibbs measure (TIGM); Let $$k=2$$ and $$\Lambda_{0}<+\infty$$ . For the model under constraint such that at $$G$$ -admissible graph the loops are imposed at two vertices of the graph, the uniqueness of TIGM is proved; Let $$k=2$$ and $$\Lambda_{0}<+\infty$$ . For the model under constraint such that at $$G$$ -admissible graph the loops are imposed at three vertices of the graph, the uniqueness and non-uniqueness conditions of TIGMs are found.

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