GIBBS MEASURES FOR THE HC BLUME-CAPEL MODEL WITH COUNTABLY MANY STATES ON A CAYLEY TREE

We study the Blume-Capel model with a countable set Z of spin values and a force J is an element of R of interaction between the nearest neighbors on a Cayley tree of order k >= 2. The following results are obtained. Let theta = e(-J/T) , T > 0, be the temperature. For theta >= 1, there exist no translation invariant Gibbs measures or 2-periodic Gibbs measures. For 0 < theta < 1, we prove the uniqueness of a translation-invariant Gibbs measure. Let circle minus = Sigma(i)theta((k+1)i2) and circle minus(cr )(k) = k(k)(k - 1)(k+1). If 0 < circle minus < circle minus(cr) then there exists exactly one 2-periodic Gibbs measure that is translation invariant. For circle minus > circle minus(cr), there exist exactly three 2-periodic Gibbs measures, one of which is a translation-invariant Gibbs measure.