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We consider ferromagnetic long-range Ising models which display phasetransitions. They are long-range one-dimensional Ising ferromagnets, in whichthe 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 isvia a kind of Peierls contour argument, using the adaptation of theFr\"ohlich-Spencer contours for $\alpha \neq 0$, 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 $\alpha_+=\log(3)/\log(2)-1$, although in theliterature dealing with contour methods for these models it is generallyassumed that $J(1)\gg1$, we can show that this condition can be removed in thecontour analysis. In addition, combining our theorem with a recent result ofLittin and Picco we prove the persistence of the contour proof of the phasetransition for any $\alpha \in [0,1)$. Moreover, we show that when we add amagnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-\gamma}$ and$\gamma >\max\{1-\alpha, 1-\alpha^* \}$ where $\alpha^*\approx 0.2714$, thetransition still persists.

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