Translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree

Abstract: In this paper we consider translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree of order k. An approximate critical temperature T_{cr} is found such that for T\geq T_{cr} there exists a unique translation-invariant Gibbs measure and for 0<T<T_{cr} there are exactly three translation-invariant Gibbs measures. In addition, we studied the problem of (not) extremality for the unique Gibbs measure.