Gibbs measures for the HC-Blum—Kapel model with a countable number of states on the Cayley tree

The report is devoted to the study of the Gibbs measure for the HC-Blum—Kapel model with a countable set of $\mathbb{Z}$ spin values and the interaction strength $J\in \mathbb{R}$ of nearest neighbors on the Cayley tree of order $k\ge 2 $. Conditions for the existence and non-existence of a translation-invariant Gibbs measure are found and the uniqueness of such a measure is proved under the condition of existence. In addition, for the model under consideration, periodic Gibbs measures with period two are studied and the exact value of the parameter ${{\Theta }_{cr}}$ is found such that for $0<\Theta \le {{\Theta }_{cr}} $ there is exactly one periodic Gibbs measure ${{\mu }_{0}}$, which is translation-invariant, and for $\Theta >{{\Theta }_{cr}}$ there are exactly three periodic Gibbs measures $ {{\mu }_{0}},\,\,{{\mu }_{1}},\,\,{{\mu }_{2}}$, where the measures ${{\mu } _{1}},\,\,{{\mu }_{2}}$ are periodic (non-translation invariant) Gibbs measures with period two.