On the validity of the van der Waals theory in Ising systems with long range interactions

We consider an Ising system in $d \ge 2$ dimensions with ferromagnetic spin-spin interactions $-J_\g(x,y)\s(x)\s(y)$, $x$, $y \in \Bbb Z^d$, where $J_\g(x,y)$ scales like a Kac potential. We prove that when the temperature is below the mean field critical value, for any $\g$ small enough (i.e. when the range of the interaction is long but finite), there are only two pure homogeneous phases, as stated by the van der Waals theory. After introducing block spin variables and relying on the Peierls estimates proved in [\rcite{CP}], the proof follows that in [\rcite{GM}] on the translationally invariant states at low temperatures for nearest neighbor interactions, supplemented by a ``relative uniqueness criterion for Gibbs fields" which yields uniqueness in a restricted ensemble of measures, in a context where there is a phase transition. This criterion is derived by introducing special couplings as in [\rcite {BM}] which reduce the proof of relative uniqueness to the absence of percolation of ``bad event