Highly Anisotropic Scaling Limits

We consider a highly anisotropic \(d=2\) Ising spin model whose precise definition can be found at the beginning of Sect. 2. In this model the spins on a same horizontal line (layer) interact via a \(d=1\) Kac potential while the vertical interaction is between nearest neighbors, both interactions being ferromagnetic. The temperature is set equal to 1 which is the mean field critical value, so that the mean field limit for the Kac potential alone does not have a spontaneous magnetization. We compute the phase diagram of the full system in the Lebowitz–Penrose limit showing that due to the vertical interaction it has a spontaneous magnetization. The result is not covered by the Lebowitz–Penrose theory because our Kac potential has support on regions of positive codimension.