We consider the Potts model with three spin values and with competing interactions of radius r = 2 on the Cayley tree of order k = 2. We completely describe the ground states of this model and use the contour method on the tree to prove that this model has three Gibbs measures at sufficiently low temperatures.
In the present paper we continue the investigation from Bovier and Külske (1996 J. Stat. Phys. 83 751-59) and consider the SOS (solid-on-solid) model on the Cayley tree of order k 2. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.
Abstract. In the paper we generalize results of paper [12] for a q-component models on a Cayley tree of order k ≥ 2. We generalize them in two directions: (1) from k = 2 to any k ≥ 2; (2) from concrete examples (Potts and SOS models) of q− component models to any qcomponent models (with nearest neighbor interactions). We give a set of periodic ground states for the model. Using the contour argument which was developed in [12] we show existence of q different Gibbs measures for q-component models on Cayley tree of order k ≥ 2.
Mathematics Subject Classifications (2000). 82B05, 82B20, 60K35, 05C05.
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