Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree

Abstract: In this paper, we consider the Potts-SOS model where the spin takes values in the set ${0, 1, 2}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions. Moreover, we investigate whether these Gibbs measures are extremal or non-extremal in the set of all Gibbs measures.