Boundary configurations for Gibbs measures of some models defined on lattices

In this talk, we shall begin by recall essential definitions and concepts from the theory of Gibbs measures defined on lattices. We shall then present results regarding the set of translation-invariant Gibbs splitting measures for the three-state SOS model with an external field on the Cayley tree of arbitrary order. Next, we shall discuss the existence of a non-translation-invariant two-periodic Gibbs measure for the three-state SOS model with the translation-invariant and two-periodic external fields on the Cayley tree of arbitrary order. Finally, we give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with the translation-invariant splitting Gibbs measures.