  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.
