Absence of Dobrushin States for 2d Long-Range Ising Models
Loren Coquille (Université Grenoble Alpes)
A.C.D. van Enter (Rijksuniversiteit Groningen)
Arnaud Le Ny (Université Paris-Est, Eindhoven University of Technology)
W.M. Ruszel (TU Delft - Applied Probability)
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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.