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 intensity λ ≥ 0 and scaling parameter λ > 0. In the Lebowitz Penrose limit γ → 0⁺ the phase diagram in the (T, λ) positive quadrant is described by a critical curve λ^mf(T), which separates the regions with one and two phases, respectively below and above the curve. We prove that if λ > λ^mf(T), i.e. above the curve, there are at least two Gibbs states for small values of γ. If instead λ < λ^mf(T) and if the reference Gibbs state (i.e. without the Kac potential) satisfies a mixing condition at the temperature T, then, at the same temperature the full interaction (i.e. with also the Kac potential) satisfies the Dobrushin Shlosman uniqueness condition for small values of γ so that there is a unique Gibbs state.