In this paper, we study the translation-invariant Gibbs measures for the fertile three-state Hard-Core (HC) models with the activity parameter λᵢ>0, i=0,1,2 on a Cayley tree. It is known that there exist four types of such models: "wrench", "wand", "hinge" and "pipe". In the case "hinge" under some conditions on parameters on a Cayley tree of order two, three the exact number of the translation invariant Gibbs measures found.
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N. Mutalliyev (2024) studied this question.
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