We consider the phase separation problem for the one--dimensional ferromagnetic Ising model with long--range two--body interaction, $J(n)=n^{ -2+\a}$ where $n\in \N$ denotes the distance of the two spins and $ \alpha \in ]0,\a_+[$ with $\a_+=(\log 3)/(\log 2) -1$. 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 $\frac{ 1}{ 2}(1-\frac{ m}{ m_\b})$ of the volume with the minus phase inside and the rest of the volume is the plus phase, here $m_\b>0 $ is the spontaneous magnetization.
Categories: math-ph math.MP