Abstract
AbstractWe consider—in a uniformly strictly convex potential regime—two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters through the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite volume in dimension $$d = 2$$ d = 2 , while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn (Commun Math Phys 185:1–36, 1997). Van Enter and Külske proved in (Ann Appl Probab 18(1):109–119, 2008) that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in $$d = 2$$ d = 2 . In Cotar and Külske (Ann Appl Probab 22(5):1650–1692, 2012) we proved the existence of shift-covariant random gradient Gibbs measures for model (A) when $$d\ge 3$$ d ≥ 3 , the disorder is i.i.d and has mean zero, and for model (B) when $$d\ge 1$$ d ≥ 1 and the disorder has a stationary distribution. In the present paper, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt$$u\in {\mathbb R}^{d}$$ u ∈ R d and with the corresponding annealed measure being ergodic: for model (A) when $$d\ge 3$$ d ≥ 3 and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when $$d\ge 1$$ d ≥ 1 and for any stationary disorder-dependence structure. We also compute for both models for any gradient Gibbs measure constructed as in Cotar and Külske (Ann Appl Probab 22(5):1650–1692, 2012), when the disorder is i.i.d. and its distribution satisfies a Poincaré inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure.
Highlights
Phase separation in Rd+1 can be described by effective interface models for the study of phase boundaries at a mesoscopic level in statistical mechanics
There are striking examples where disorder acts in an opposite way: Non-uniqueness of the Gibbs measure and a new type of ordering can even be created by the introduction of quenched randomness of a random field type
Such an order-bydisorder mechanism was proved to happen in the rotator model in the presence of a uniaxial random field, see [16] and [17]
Summary
Phase separation in Rd+1 can be described by effective interface models for the study of phase boundaries at a mesoscopic level in statistical mechanics. There are striking examples where disorder acts in an opposite way: Non-uniqueness of the Gibbs measure and a new type of ordering can even be created by the introduction of quenched randomness of a random field type. Such an order-bydisorder mechanism was proved to happen in the rotator model in the presence of a uniaxial random field, see [16] and [17]. For our second main result for both models A and B, we will work under the following slightly more restrictive Poincaré inequality assumption on the distribution γ of the disorder ξ(0), The rest of the introduction is structured as follows: in Sect. 1.2 we define in detail the notions of finite-volume and infinite-volume (gradient) Gibbs measures for model A, in Sect. 1.3 we sketch the corresponding notions for model B, and in Sect. 1.4 we present our main results and their connection to the existing literature
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