Layered Systems at the Mean Field Critical Temperature

We consider the Ising model on \(\mathbb Z\times \mathbb Z\) where on each horizontal line \(\{(x,i), x\in \mathbb Z\}\) the interaction is given by a ferromagnetic Kac potential with coupling strength \(J_{ \gamma }(x,y)\sim \gamma J(\gamma (x-y))\) at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength \(\epsilon \) and prove that for every \(\epsilon >0\) the systems exhibits phase transition provided \(\gamma >0\) is small enough.