Nonexistence of random gradient Gibbs measures in continuous interface models in d=2

We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension d=2, while there are .gradient Gibbs measures. describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in d=2. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In d=3 where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.