Absence of Dobrushin states for $2d$ long-range Ising models

TL;DR

This paper proves that for the 2D long-range Ising model with decay exponent greater than 2, extremal non-translation-invariant Gibbs states known as Dobrushin states do not exist, highlighting differences from short-range models.

Contribution

The paper establishes the non-existence of Dobrushin states in 2D long-range Ising models with decay exponent > 2, extending understanding of phase behavior in long-range interactions.

Findings

Dobrushin states do not exist for the specified long-range Ising models.

The result applies when the decay exponent α > 2.

Discussion of potential extensions and related phenomena.

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 $J_{xy} \geq 0$. 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.