Continuous interfaces with disorder: Even strong pinning is too weak in two dimensions

We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta pinning of strength ε$\epsilon$ at height zero. There is a detailed mathematical understanding of the depinning transition in two dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for ε>0$\epsilon >0$ and diverges like |logε|$|log\epsilon |$ for ε↓0$\epsilon \downarrow 0$. 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 two 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 two dimensions it implies growth faster than that of 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 ε↑∞$\epsilon \uparrow \infty$.