A HC model with countable set of spin values: uncountable set of Gibbs measures

Abstract

We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters ${\lambda }_i>0,i\in \mathbb{Z}\setminus \{0\}$. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.

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• Fixed points of an infinite dimensional operator related to Gibbs measures