# Phase separation for the long range one--dimensional ising model

Research paper by **Marzio Cassandro, Immacolata Merola, Pierre Picco**

Indexed on: **07 Nov '16**Published on: **07 Nov '16**Published in: **arXiv - Mathematical Physics**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

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.