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

Abstract

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.

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Acknowledgements

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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Correspondence to Loren Coquille.

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Coquille, L., van Enter, A.C.D., Le Ny, A. et al. Absence of Dobrushin States for 2d Long-Range Ising Models. J Stat Phys 172, 1210–1222 (2018). https://doi.org/10.1007/s10955-018-2097-7

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  • DOI: https://doi.org/10.1007/s10955-018-2097-7

Keywords

  • Gibbs states
  • Long-range Ising model
  • Dobrushin states
  • Interface fluctuations

Mathematics Subject Classification

  • 82B05
  • 82B20
  • 82B26