Gibbs measures for fertile models with hard-core interactions and four states

We consider fertile models with hard interactions, four states, and an activity parameter $\lambda>0$ on a Cayley tree. We show that there are three types of such models: “stick,” “key,” and “generalized key.” For the “generalized key” model on a Cayley tree of order $k=4$, the uniqueness of the translation-invariant Gibbs measure is proved, and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones are found. Moreover, in the case of a fertile graph of the “stick” type, the translation invariance of double-periodic Gibbs measures on a Cayley tree of orders $k=2,3,4$ is shown and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones on a Cayley tree of order $k\geq5$ are found.