\(p\)-Adic Quasi Gibbs Measures for the Three-State SOS Model with an External Field on a Cayley Tree

We consider a three-state solid-on-solid (SOS) model in the presence of a nonzero external field on a Cayley tree. A system of functional equations corresponding to this model is derived, where each solution defines a quasi Gibbs measure. Based on this system, we investigate the translation-invariant \(p\) -adic quasi Gibbs measures (TIpQGMs) of the model. In the case of a binary Cayley tree, we explicitly determine the TIpQGMs under the same conditions. Furthermore, by analyzing the boundedness of these measures, we establish the existence of a phase transition.