We consider ferromagnetic long-range Ising models which display phase transitions. They are one-dimensional Ising ferromagnets, in which the interaction is given by Jx,y=J(|x-y|)≡1|x-y|2-α with α∈ [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 Fröhlich–Spencer contours for α≠ 0 , proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α= 0 and conjectured by Cassandro et al for the region they could treat, α∈ (0 , α+) for α+= log (3) / log (2) - 1 , although in the literature dealing with contour methods for these models it is generally assumed that J(1) ≫ 1 , we will 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 α∈ [0 , 1). Moreover, we show that when we add a magnetic field decaying to zero, given by hx= h∗· (1 + | x|) -γ and γ> max { 1 - α, 1 - α∗} where α∗≈ 0.2714 , the transition still persists.
CITATION STYLE
Bissacot, R., Endo, E. O., van Enter, A. C. D., Kimura, B., & Ruszel, W. M. (2018). Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields. Annales Henri Poincare, 19(8), 2557–2574. https://doi.org/10.1007/s00023-018-0693-3
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