Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree

Abstract

In this paper, we study the HC-model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k\ge 2$. 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 is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);

• For $\Lambda <+\infty$, the uniqueness of TIGM is proved;

• Let $\Lambda _\textrm{cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<\Lambda \le \Lambda _\textrm{cr}$, then there is exactly one TPGM that is TIGM;

• For $\Lambda >\Lambda _\textrm{cr}$, there are exactly three TPGMs, one of which is TIGM.