Highly anisotropic scaling limits

TL;DR

This paper analyzes a highly anisotropic 2D Ising model with mixed interactions, revealing spontaneous magnetization due to vertical coupling, despite the mean field critical temperature, and extends beyond traditional Lebowitz-Penrose theory.

Contribution

It computes the phase diagram of an anisotropic Ising model with non-standard Kac potential support, demonstrating spontaneous magnetization beyond classical mean field predictions.

Findings

Vertical interaction induces spontaneous magnetization.

Phase diagram computed in Lebowitz-Penrose limit.

Kac potential support on positive codimension regions.

Abstract

We consider a highly anisotropic $d=2$ Ising spin model whose precise definition can be found at the beginning of Section 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.