Tranlation-invariant Gibbs measures for the Hard-Core model with a countable set of spin values

In this paper, we study the Hard Core (HC) model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k=2$. This model is defined by a countable set of parameters (that is, the activity function $\lambda_i>0$, $i\in \mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: Let $k\geq 2$ and $\Lambda=\sum_i\lambda_i$. For $\Lambda=+\infty$ there is no translation-invariant Gibbs measure (TIGM); Let $k=2$ and $\Lambda<+\infty$. For the model under constraint such that at $G$-admissible graph the loops are imposed at two vertices of the graph, the uniqueness of TIGM is proved; Let $k=2$ and $\Lambda<+\infty$. For the model under constraint such that at $G$-admissible graph the loops are imposed at three vertices of the graph, the uniqueness and non-uniqueness conditions of TIGMs are found.Comment: arXiv admin note: text overlap with arXiv:2206.06333 by other author

New class of Gibbs measures for two state Hard-Core model on a Cayley tree

In this paper, we consider a Hard-Core $(HC)$ model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model

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 $k\geq 2$, 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 $k\geq 2$.Comment: 19 pages, 2 figure

Gibbs measures for HC-model with a countable set of spin values on a Cayley tree

In this paper, we study the HC-model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k\geq 2$. This model is defined by a countable set of parameters (that is, the activity function $\lambda_i>0$, $i\in \mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: - Let $\Lambda=\sum_i\lambda_i$. For $\Lambda=+\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $\Lambda<+\infty$, the uniqueness of TIGM is proved; - Let $\Lambda_{\rm cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<\Lambda\leq\Lambda_{\rm cr}$, then there is exactly one TPGM that is TIGM; - For $\Lambda>\Lambda_{\rm cr}$, there are exactly three TPGMs, one of which is TIGM.Comment: 15 pages, 1 figur