Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order $$k=3$$

We study fertile hard-core models with three states and an activity parameter $$\lambda>0$$ on the Cayley tree of order $$k=3$$ . It is known that there are four types of such models. For two of them, we find the regions where the unique translation-invariant Gibbs measure is (not) extremal. For one of the considered models, we find the conditions under which the extremal measure is not unique.

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