Gibbs measures for the three-state SOS model with external field on a Cayley tree

We consider a three-state solid-on-solid (SOS) model in the presence of nonzero external field on a Cayley tree. We obtain a system of functional equations for this model, which each positive solution of the system corresponds to a splitting Gibbs measure. Using this, we study the translation-invariant splitting Gibbs measures of the model. On the binary tree, we find all the translation-invariant splitting Gibbs measures of the model.