One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point

Abstract

We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long-range two-body interaction, J(n) = n −2+α , where $${n\in {\rm {I\!N}}}$$ denotes the distance of the two spins and $${\alpha \in [0,\alpha_+[}$$ with α + = (log 3)/(log 2) −1. We prove that when α = 0 the localization of the phase separation fluctuates macroscopically with a non-uniform explicit limiting law, while when 0 < α < α + the macroscopic fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently scaled. The mean magnetization profile is also given.