Weakly periodic gibbs measures for the HC model with a countable set of spin values

In this paper, we study the weakly periodic (nonperiodic) Gibbs measures for the Hard Core (HC) model with a countable set ℤ of spin values and with a countable set of parameters λ_i > 0, i ∈ ℤ, on a Cayley tree of order k ≥ 2. For the considered model in the case ∑_i λ_i < +∞, a complete description of weakly periodic Gibbs measures is obtained for any normal divisor of index two and in the case ∑_i; λ_i = +∞, it is shown that there is no weakly periodic Gibbs measure. Moreover, in the case of a normal divisor of index four the uniqueness conditions for weakly periodic Gibbs measures are found. Further, under certain conditions an exact critical value is found that ensures the existence of weakly periodic Gibbs measures.