Nonprobability Gibbs measures for the HC model with a countable set of spin values for a “wand”-type graph on a Cayley tree

We study Gibbs measures for the HC model with a countable set $\mathbb Z$ of spin values and a countable set of parameters (i.e., with the activity function $\lambda_i>0$, $i\in \mathbb Z$) in the case of a “wand”-type graph. In this case, analyzing a functional equation that ensures the consistency condition for finite-dimensional Gibbs measures, we obtain the following results. Exact values of the parameter $\lambda_{\mathrm{cr}}$ are determined; it is shown that for $0<\lambda\leq\lambda_{\mathrm{cr}}$, there exists exactly one translation-invariant nonprobabilistic Gibbs measure, and for $\lambda>\lambda_{\mathrm{cr}}$, there exist precisely three such measures on a Cayley tree of order $2$, $3$, or $4$. We obtain the uniqueness conditions for $2$-periodic nonprobabilistic Gibbs measures on a Cayley tree of an arbitrary order, as well as exact values of the parameter $\lambda_{\mathrm{cr}}$; we also show that for $\lambda\geq\lambda_{\mathrm{cr}}$, there exists precisely one such a measure, and for $0<\lambda<\lambda_{\mathrm{cr}}$, there exist precisely three such measures on a Cayley tree of order $2$ or $3$.