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

Journal article (2018)

Authors

Loren Coquille Université Grenoble Alpes
A.C.D. van Enter Rijksuniversiteit Groningen
Arnaud Le Ny Eindhoven University of Technology, Université Paris-Est
W.M. Ruszel Applied Probability - EEMCS
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2018 Loren Coquille, A.C.D. van Enter, Arnaud Le Ny, W.M. Ruszel
DOI: https://doi.org/10.1007/s10955-018-2097-7
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Published Date

2018

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

Abstract

We consider the two-dimensional Ising model with long-range pair interactions of the form (Formula presented.) with (Formula presented.), mostly when (Formula presented.). 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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