Phase Transitions in Layered Systems

Abstract

We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, 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 $A\ge 2$ 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.