Critical behavior of the massless free field at the depinning transition

We consider the d-dimensional massless free field localized by a delta-pinning of strength e. We study the asymptotics of the variance of the field, and of the decay-rate of its 2-point function, as e goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse and longitudinal correlation lengths of the corresponding d+1-dimensional effective interface model in a non-mean-field regime. We also describe the set of pinned sites at small e, for a broad class of d-dimensional massless models.