Abstract
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.
