PHASE-DIAGRAM OF ISING SYSTEMS WITH ADDITIONAL LONG-RANGE FORCES

We consider ferromagnetic Ising systems where the interaction is given by the sum of a fixed reference potential and a Kac potential of inte nsity lambda greater than or equal to 0 and scaling parameter gamma > 0. In the Lebowitz Penrose limit gamma --> 0(+) the phase diagram in t he (T;lambda) positive quadrant is described by a critical curve lambd a(mf)(T), which separates the regions with one and two phases, respect ively below and above the curve. We prove that if lambda > lambda(mf)( T), i.e, above the curve, there are at least two Gibbs states for smal l values of gamma, If instead lambda < lambda(mf)(T) and if the refere nce Gibbs state (i,e, without the Kac potential) satisfies a mixing co ndition at the temperature T, then, at the same temperature the full i nteraction (i,e. with also the Kac potential) satisfies the Dobrushin Shlosman uniqueness condition for small values of gamma so that there is a unique Gibbs state.