Nonprobability Gibbs measures for the HC model with a countable set of spin values for a “wand”-type graph on a Cayley tree
Аннотация
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 $$3$$ .