Gibbs measures for HC-model with a countable set of spin values on a Cayley tree

In this paper, we study the HC-model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k\geq 2$. 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 $\Lambda=\sum_i\lambda_i$. For $\Lambda=+\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $\Lambda<+\infty$, the uniqueness of TIGM is proved; - Let $\Lambda_{\rm cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<\Lambda\leq\Lambda_{\rm cr}$, then there is exactly one TPGM that is TIGM; - For $\Lambda>\Lambda_{\rm cr}$, there are exactly three TPGMs, one of which is TIGM.