Gibbs Measures for HC-Model with a Cuountable 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 \(k\ge 2\). 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.