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$\alpha \in [0, 1)$,inparticular,$J(1)=1$. 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 Fr"ohlich-Spencer contours for $\alpha \mathrm{\ne }0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr"ohlich and Spencer for $\alpha =0$ and conjectured by Cassandro et al for the region they could treat, $\alpha \in (0,{\alpha }_{+})$ for ${\alpha }_{+}=\log (3)\mathrm{/}\log (2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg 1$, 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 $h_x=h_{\ast }\cdot (1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\gamma }$ and $\gamma >\max \{1-\alpha ,1-{\alpha }^{\ast }\}$ where ${\alpha }^{\ast }\approx 0.2714$, the transition still persists.