Gibbs measures for a Hard-Core model with a countable set of states

In this paper, we focus on studying non-probability Gibbs measures for a Hard Core (HC) model on a Cayley tree of order $k\ge 2$, where the set of integers $\mathbb{Z}$ is the set of spin values. It is well-known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalisable boundary law corresponds to a Gibbs measure. However, a non-normalisable boundary law can define gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC-model on a Cayley tree of any order $k\ge 2$.