Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields

Inspired by Fr\"{o}hlich-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 $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on a multi-scale 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 $h^*|x|^{-\delta}$, where $h^* >0$. For $d<\alpha<d+1$, the phase transition occurs when $\delta>\alpha-d$, and when $h^*$ is small enough over the critical line $\delta=\alpha-d$. For $\alpha \geq d+1$, $\delta>1$ is enough to prove the phase transition, and for $\delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.