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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 Fr\\\"ohlich-Spencer contours for $\\alpha \\neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. 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D. van Enter, Bruno Kimura, Eric O. Endo, Rodrigo Bissacot, Wioletta M. Ruszel","submitted_at":"2017-10-09T08:28:13Z","abstract_excerpt":"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 Fr\\\"ohlich-Spencer contours for $\\alpha \\neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. 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