Annales Henri Poincaré

, Volume 19, Issue 8, pp 2557–2574 | Cite as

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

  • Rodrigo BissacotEmail author
  • Eric O. Endo
  • Aernout C. D. van Enter
  • Bruno Kimura
  • Wioletta M. Ruszel


We consider ferromagnetic long-range Ising models which display phase transitions. They are 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 Fröhlich–Spencer contours for \(\alpha \ne 0\), proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for \(\alpha =0\) and conjectured by Cassandro et al for the region they could treat, \(\alpha \in (0,\alpha _{+})\) for \(\alpha _+=\log (3)/\log (2)-1\), although in the literature dealing with contour methods for these models it is generally assumed that \(J(1)\gg 1\), 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 \(\alpha \in [0,1)\). Moreover, we show that when we add a magnetic field decaying to zero, given by \(h_x= h_*\cdot (1+|x|)^{-\gamma }\) and \(\gamma >\max \{1-\alpha , 1-\alpha ^* \}\) where \(\alpha ^*\approx 0.2714\), the transition still persists.


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We thank the referee for a number of helpful remarks. RB and EE thank Jorge Littin for providing us with his thesis and with a preliminary version of [29] and for fruitful discussions. RB thanks Maria Eulália Vares for calling his attention to the problem of getting rid of the condition \(J(1)\gg 1\) when one uses contours for Dyson models. We thank Arnaud Le Ny, Marzio Cassandro and Luiz Renato Fontes for all they taught in earlier collaborations and/or in discussions and helpful suggestions on the manuscript. EE is supported by FAPESP Grants 2014/10637-9 and 2015/14434-8. RB is supported by FAPESP Grants 2016/25053-8, 2016/08518-7 and CNPq Grants 312112/2015-7 and 446658/2014-6.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Rodrigo Bissacot
    • 1
    Email author
  • Eric O. Endo
    • 1
    • 2
  • Aernout C. D. van Enter
    • 2
  • Bruno Kimura
    • 3
  • Wioletta M. Ruszel
    • 3
  1. 1.Institute of Mathematics and Statistics (IME-USP)University of São PauloSão PauloBrazil
  2. 2.Johann Bernoulli InstituteUniversity of GroningenGroningenThe Netherlands
  3. 3.Delft Institute for Applied MathematicsTechnical University DelftDelftthe Netherlands

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