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تسجيل مستخدم جديدLong-Range Ising Models: Contours, Phase Transitions and Decaying Fields
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Inspired by Fr{o}hlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $mathbb{Z}^d$, $dgeq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $alpha>d$ and improves previous 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 coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial 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 the phase transition, and for $delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.
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