Hard-Core Model with Countable States: Translation-Invariant Gibbs Measures

In this paper, we investigate the translation-invariant Gibbs measures (TIGM) for the Hard-Core (HC) lattice model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree of order $k \geq 2$. This statistical mechanics model is uniquely characterized by an infinite set of parameters governed by a~positive activity function $\lambda_{i}>0$, $i \in \mathbb{Z}$. We explicitly derive the functional consistency equation and establish the exact normalizability conditions required for the boundary laws of the system. The analysis specifically focuses on the convergence of the activity series partitions, denoted by $\sum_{j\in\mathbb{Z}_{0}}\lambda_{2j}=\Delta$ and ${\sum_{j\in\mathbb{Z}}\lambda_{2j+1}=\Psi}$. In the symmetric subcase where $\Delta=\Psi\equiv\gamma$, we determine the precise critical value $\gamma_{cr}(k)$, proving that a unique TIGM exists for $\gamma \le \gamma_{cr}(k)$, while there exist exactly three distinct TIGMs when $\gamma>\gamma_{cr}(k)$ on Cayley trees of orders $k=2,3,4,5$. Furthermore, a generalized lower bound for the total number of TIGMs is formulated for all orders $k \geq 2$. Finally, in the asymmetric subcase ($\Delta \neq \Psi$), we provide a comprehensive mathematical description and classification of all translation-invariant Gibbs measures on Cayley trees of orders $k=1,2,3$.

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