Existence of random gradient states

We consider two versions of random gradient models. 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. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are “gradient Gibbs measures” describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and K"{u}lske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt $u\in $\mathbb{R}$^d$formodelAwhen$d\geq3$ and the disorder has mean zero, and for model B when $d\ge 1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge 3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.