Non-existence 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 others. 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 non-existence 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.