Contour methods for Dyson models: Weakening nearest-neighbour interactions and adding decaying fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are one-dimensional Ising ferromagnets, in which the interaction is given by Jx,y=J(|x−y|)≡1|x−y|2−α with α∈[0,1) , in particular, J(1)=1 . For this class of models, one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich–Spencer contours for α≠0 , proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α=0 and conjectured by Cassandro et al for the region they could treat, α∈(0,α+) for α+=log(3)/log(2)−1 , although in the literature dealing with contour methods for these models it is generally assumed that J(1)≫1 , we will show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α∈[0,1) . Moreover, we show that when we add a magnetic field decaying to zero, given by hx=h∗⋅(1+|x|)−γ and γ>max{1−α,1−α∗} where α∗≈0.2714 , the transition still persists.