Uniqueness of gradient Gibbs measures with disorder

We 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.