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 multi-scale analysis, works for the sharp region $\hat{I}\pm >d$ and improves previous results obtained by Park for $\hat{I}\pm >3d+1$, and by Ginibre, Grossmann, and Ruelle for $\hat{I}\pm >d+1$, where $\hat{I}\pm$ 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 $\hat{I}´>0$ as $h^{\ast }|x{|}^{-\hat{I}´}$, where $h^{\ast }>0$. For $d<\hat{I}\pm <d+1$, the phase transition occurs when $\hat{I}´>\hat{I}\pm -d$, and when $h^{\ast }$ is small enough over the critical line $\hat{I}´=\hat{I}\pm -d$. For $\hat{I}\pm \ge d+1$, $\hat{I}´>1$ is enough to prove the phase transition, and for $\hat{I}´=1$ 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.

29 pages, 6 figures. Final version accepted for publication on JEMS