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 $λ_i>0$, $i\in \mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: - Let $Λ=\sum_iλ_i$. For $Λ=+\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $Λ<+\infty$, the uniqueness of TIGM is proved; - Let $Λ_{\rm cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<Λ\leqΛ_{\rm cr}$, then there is exactly one TPGM that is TIGM; - For $Λ>Λ_{\rm cr}$, there are exactly three TPGMs, one of which is TIGM.

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