Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

Rodrigo Bissacot, Eric O. Endo, Aernout C.D. van Enter, Bruno Kimura, Wioletta M. Ruszel

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)

Abstract

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 +

Original languageEnglish
Pages (from-to)2557-2574
Number of pages18
JournalAnnales Henri Poincare
Volume19
Issue number8
DOIs
Publication statusPublished - 2018

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