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

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 ∈ N denotes the distance of the two spins and α ∈ [0, α+[ 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.