Upper Bounds on the Critical Temperature for Kac Potentials

Abstract

We consider an Ising system in two dimensions with a two body ferromagnetic interaction Jγ(x, y) that depends on the Kac scaling parameter γ. We prove that the inverse critical temperature βcr(γ) is strictly above the mean-field value (equal to 1), namely that there exists C>0 so that for any b 1 + bγ2log γ−1 for all γ sufficiently small. The temperature shift Cγ2log γ−1 is to leading orders equal to the covariance of the magnetization fluctuations.