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 $\pm$-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.

revised version