Journal of Statistical Physics

, Volume 172, Issue 5, pp 1210–1222 | Cite as

Absence of Dobrushin States for 2d Long-Range Ising Models

  • Loren CoquilleEmail author
  • Aernout C. D. van Enter
  • Arnaud Le Ny
  • Wioletta M. Ruszel


We consider the two-dimensional Ising model with long-range pair interactions of the form \(J_{xy}\sim |x-y|^{-\alpha }\) with \(\alpha >2\), mostly when \(J_{xy} \ge 0\). We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed ± boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman–Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.


Gibbs states Long-range Ising model Dobrushin states Interface fluctuations 

Mathematics Subject Classification

82B05 82B20 82B26 



LC and ALN have been partially supported by the CNRS PEPS Project “Ising” and visitor Grants by “Networks” and “European Women in Math”. ALN has benefited from the support of CNRS, Eurandom UMI CNRS 3022, and the Dutch Gravitation Grant “Networks”. We would like to thank Yvan Velenik for his comments and encouragements.


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Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, CNRS, Institut FourierGrenobleFrance
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  3. 3.LAMA UMR CNRS 8050, UPEC, Université Paris-EstCréteil CedexFrance
  4. 4.Eurandom, TU/e EindhovenEindhovenThe Netherlands
  5. 5.Delft Institute of Applied MathematicsTechnical University DelftDelftThe Netherlands

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