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

In this paper, we focus on studying the non-probability Gibbs measures for a Hard-Core (HC) model on a Cayley tree of order [Formula: see text], where the set of integers [Formula: see text] 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 (with strictly positive coordinates) defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalizable boundary law corresponds to a Gibbs measure. However, a non-normalizable boundary law can define the 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 [Formula: see text].

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