Phase Transitions in Layered Systems

We consider the Ising model on (mathbb Ztimes mathbb Z) where on each horizontal line ({(x,i), xin mathbb Z}) , called “layer”, the interaction is given by a ferromagnetic Kac potential with coupling strength (J_{ gamma }(x,y)={ gamma }J({ gamma }(x-y))) , where (J(cdot )) is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength ({ gamma }^{A}) , where (Age 2) is fixed, and prove that for any (beta ) larger than the mean field critical value there is a phase transition for all ({ gamma }) small enough.