In the paper, we consider the λ-model with spin values {1, 2, 3} on the Cayley tree of order two. We first describe ground states of the model. Moreover, we also proved the existence of translation-invariant Gibb measures for the λ-model which yield the existence of the phase transition. Lastly, we established the exitance of 2-periodic Gibbs measures for the model.

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In this paper we consider the λ-model on the Cayley tree of order two. We describe periodic and weakly periodic ground states for the considered model.

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Abstract.We investigate an Ising model with two restricted competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order 5. The translation Gibbs measures is considered for this model. Our result of the critical curve shows that the phase transition occurs in this model, further it confirms a particular case of a conjecture.