Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)equiv frac{1}{|x-y|^{2-alpha}}$ with $alpha in [0, 1)$, 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 Frohlich-Spencer contours for $alpha eq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Frohlich and Spencer for $alpha=0$ and conjectured by Cassandro et al for the region they could treat, $alpha in (0,alpha_{+})$ for $alpha_+=log(3)/log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)gg1$, we can 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 $alpha in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*cdot(1+|x|)^{-gamma}$ and $gamma >max{1-alpha, 1-alpha^* }$ where $alpha^*approx 0.2714$, the transition still persists.