Phase Separation for the Long Range One-dimensional Ising Model

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 given m∈] - 1 , + 1 [,if the temperature is small enough, then typical configuration for the μ+ Gibbs measure conditionally to have a empirical magnetization of the order m are made of a single interval that occupy almost a proportion 12(1-mmβ) of the volume with the minus phase inside and the rest of the volume is the plus phase, here mβ> 0 is the spontaneous magnetization.