Gibbs measures for HC-model with a countable 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 are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $\Lambda <+\infty$, the uniqueness of TIGM is proved; - Let ${\Lambda }_{\mathrm{cr}}(k)=\frac{k^k}{(k-1{)}^{k+1}}$. If $0<\Lambda \le {\Lambda }_{\mathrm{cr}}$, then there is exactly one TPGM that is TIGM; - For $\Lambda >{\Lambda }_{\mathrm{cr}}$, there are exactly three TPGMs, one of which is TIGM.

15 pages, 1 figure

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