Layered systems at the mean field critical temperature

TL;DR

This paper studies a layered Ising model at the mean field critical temperature, showing that adding a small vertical interaction induces phase transition.

Contribution

It demonstrates that even a tiny vertical interaction causes phase transition in layered systems at the mean field critical point.

Findings

Phase transition occurs for any positive vertical interaction strength.

Small horizontal interaction parameter fects phase transition.

Layered systems exhibit phase transition at critical temperature with added vertical coupling.

Abstract

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.