Long-range Ising models: Contours, phase transitions and decaying fields

Abstract

Inspired by Fröhlich–Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on ${\mathbb{Z}}^d$, $d\ge 2$. The argument, which is based on a multiscale analysis, works for the sharp region $\alpha >d$ and improves previous results obtained by Park for $\alpha >3d+1$, and by Ginibre, Grossmann and Ruelle for $\alpha >d+1$, where $\alpha$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomially decaying magnetic field with power $\delta >0$ as $h^{\ast }|x{|}^{-\delta }$, where $h^{\ast }>0$. For $d<\alpha <d+1$, the phase transition occurs when $\delta >\alpha -d$, and when $h^{\ast }$ is small enough over the critical line $\delta =\alpha -d$. For $\alpha \ge d+1$, $\delta >1$ is enough to prove the phase transition, and for $\delta =1$ we have to ask $h^{\ast }$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.