Continuous interfaces with disorder: even strong pinning is too weak in 2 dimensions

We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning of strength $\e$ at height zero. There is a detailed mathematical understanding of the depinning transition in $2$ dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for $\e>0$ and diverges like $|\log \e|$ for $\e\downarrow 0$. How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitrarily weak random field term is enough to beat an arbitrarily strong delta-pinning in $2$ dimensions and will cause delocalization. The proof is based on a rigorous lower bound for the overlap between local magnetizations and random fields in finite volume. In $2$ dimensions it implies growth faster than the volume which is a contradiction to localization. We also derive a simple complementary inequality which shows that in higher dimensions the fraction of pinned sites converges to one with $\e\uparrow \infty$.