GIBBS MEASURES FOR FERTILE HARD-CORE MODELS ON THE CAYLEY TREE

摘要

We study fertile hard-core models with the activity parameter lambda > 0 and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter lambda on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of lambda at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten-Stigum condition) for a measure not to be extreme.