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Inspired by Fr\"{o}hlich-Spencer and subsequent authors who introduced thenotion of contour for long-range systems, we provide a definition of contourand a direct proof for the phase transition for ferromagnetic long-range Isingmodels on $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on amulti-scale analysis, works for the sharp region $\alpha>d$ and improvesprevious 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 couplingconstant. The key idea is to avoid a large number of small contours. As anapplication, we prove the persistence of the phase transition when we add apolynomial decaying magnetic field with power $\delta>0$ as $h^*|x|^{-\delta}$,where $h^* >0$. For $d<\alpha<d+1$, the phase transition occurs when$\delta>d-\alpha$, and when $h^*$ is small enough over the critical line$\delta=d-\alpha$. For $\alpha \geq d+1$, $\delta>1$ it is enough to prove thephase transition, and for $\delta=1$ we have to ask $h^*$ small. The naturalconjecture is that this region is also sharp for the phase transition problemwhen we have a decaying field.

Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields