Abstract
In this paper, we study the HC-model with a countable set $$\mathbb Z$$ of spin values on a Cayley tree of order $$k\ge 2$$ . 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:
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