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, \(J(n)=n^{-2+\alpha }\) where \(n\in \mathbb {N}\) denotes the distance of the two spins and \( \alpha \in ]0,\alpha _{+}[\) with \(\alpha _+=(\log 3)/(\log 2) -1\). We prove that given \(m\in ]-1,+1[\), if the temperature is small enough, then typical configuration for the \(\mu ^{+}\) Gibbs measure conditionally to have a empirical magnetization of the order m are made of a single interval that occupy almost a proportion \(\frac{1}{2}(1-\frac{m}{m_\beta })\) of the volume with the minus phase inside and the rest of the volume is the plus phase, here \(m_{\beta }>0 \) is the spontaneous magnetization.