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 Jxy ∼ |x − y| −α with α > 2, mostly when Jxy ≥ 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. IntroductionWe are interested in the possible existence of so-called interface states (or Dobrushin states) for long-range Ising models in dimension d = 2. Dobrushin states are extremal infinite-volume Gibbs measures selected by mixed ±-boundary conditions originally described in [8] for the standard (nearest-neighbour) Ising model in dimension three.Depending on the question one asks, a long-range interaction can behave similarly or not to nearest-neighbour (n.n.) models. It is well-known that for n.n. interactions, interface states do not exist, see e.g. [1,16,29,24,7,9]. On the other hand there exist such extremal and non-translation-invariant Gibbs states in d ≥ 3 [2,8].In this note we consider three different examples of long-range models, with interactions which are either isotropic, long-range horizontally and n.n. vertically or bi-axial long-range (i.e. long-range in both horizontal and vertical directions with possibly different decays). By long-range interaction we mean pair interactions with coupling constants given byfor all x, y ∈ Z 2 , where | · | denotes the Euclidean norm.We first prove that for all α > 2 there are no Dobrushin states in the isotropic case, see Theorem 1. This result is similar to what happens for short-range models. However, the precise statements we can prove are weaker than what is known for n.n. models, in particular regarding the full picture of the convex set of Gibbs measures. The Aizenman-Higuchi theorem indeed states [1,24] that all infinite-volume Gibbs measures of the 2d ferromagnetic n.n. Ising model are convex combinations of the pure + and − phases. As we shall see, we have either a statement for a subset of boundary conditions, or at low enough temperatures, but we provide directions to study these problems in more generality. Note that the case of fast decays α > 3 falls within the framework of the Gertzik-Pirogov-Sinai theory. Indeed, in this decay range, a first-moment condition on the interaction applies, and allows the proof of the existence of a spontaneous magnetisation for Ising models with n.n interaction plus a long-range perturbation [21,20]. Moreover, in such a 2010 Mathematics Subject Classification. 82B05, 82B20, 82B26.

Existence of gradient Gibbs measures on regular trees which are not translation invariant

“…Definition 2. Consider the outer σ-algebra T ∇ Λ (see (8)) Then the gradient Gibbs specification is defined as the family of probability kernels…”

“…At sufficiently low temperatures they break translation invariance, see [14] and also [6]. By contrast, in low lattice dimensions d ≤ 2 such states do not exist in the Ising model, for, all Gibbs states of the Ising model are necessarily translation-invariant, see [1], [9] and also [8].…”

Markov and almost Markov properties in one, two or more directions

“…[18] and references therein). For some recent results for these long-range Ising models with polynomially decaying interactions, see [79,34,10,21,33,23,69].…”

One-Sided Versus Two-Sided Stochastic Descriptions

“…[10] and references therein). For some recent results for these long-range Dyson models with polynomially decaying interactions, see [49,21,6,12,20,14].…”