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We study fertile hard-core models with the activity parameter λ > 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 λ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of λ at which the...
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