Absence of Dobrushin States for 2<Emphasis Type="Italic">d</Emphasis> Long-Range Ising Models |
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| Authors: | Loren Coquille Aernout C D van Enter Arnaud Le Ny Wioletta M Ruszel |
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| Institution: | 1.Univ. Grenoble Alpes, CNRS, Institut Fourier,Grenoble,France;2.Johann Bernoulli Institute for Mathematics and Computer Science,University of Groningen,Groningen,The Netherlands;3.LAMA UMR CNRS 8050, UPEC, Université Paris-Est,Créteil Cedex,France;4.Eurandom, TU/e Eindhoven,Eindhoven,The Netherlands;5.Delft Institute of Applied Mathematics,Technical University Delft,Delft,The Netherlands |
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| 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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