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

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\geq 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^{*}|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.