Uniqueness and Nonuniqueness Conditions for Weakly Periodic Gibbs Measures for the Hard-core Model with a Countable Set of Spin Values

We study the Hard-Core model with a countable set $$\mathbb{Z}$$ of spin values and a countable set of parameters $$\lambda_{i}>0$$ , $$i\in\mathbb{Z}$$ on a Cayley tree. In the case the normal subgroup of index four, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters for $$\lambda<+\infty$$ , where $$\lambda=\sum_{i}\lambda_{i}$$ . Moreover, under some conditions we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures.