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 epsilon. We study the asymptotics of the variance of the field (when d = 2), and of the decay-rate of its 2-point function (when d greater than or equal to 2), as epsilon 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 epsilon, for a broad class of d-dimensional massless models.