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 $J_γ(x,y)\sim γJ(γ(x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $ε$ and prove that for every $ε>0$ the systems exhibits phase transition provided $γ>0$ is small enough.

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