Translation-invariant Gibbs measures for fertile three-state “hard core” models on a Cayley tree

We consider fertile three-state “hard core” models with the activity parameter λ > 0 on an order-three Cayley tree. It is known that there exist four types of such models: in two of them, the translation-invariant Gibbs measure is unique for λ > 0, and in the other two, a value λcr is found such that there exist only three translation-invariant Gibbs measures for λ > λcr and a single translation-invariant Gibbs measure for λ ≤ λcr.