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

TL;DR

This paper analyzes phase separation in a one-dimensional long-range ferromagnetic Ising model, showing that at low temperatures, typical configurations with a given magnetization consist of a single phase-separated interval.

Contribution

It proves that for the long-range 1D Ising model, typical low-temperature configurations with fixed magnetization form a single phase-separated interval, extending understanding of phase behavior.

Findings

Configurations with fixed magnetization are phase-separated intervals.

The size of the interval depends on the magnetization and spontaneous magnetization.

Results hold for small enough temperatures.

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.