Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)=γJ(γ(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $γ^{A}$ (where $A\ge 2$ is fixed) and prove that for any $β$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $γ$ small enough.

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