We consider the d-dimensional massless free field localized by a delta -pin
ning of strength epsilon. We study the asymptotics of the variance of the f
ield (when d = 2), and of the decay-rate of its 2-point function (when d gr
eater than or equal to 2), as epsilon goes to zero, for general Gaussian in
teractions. Physically speaking, we thus rigorously obtain the critical beh
avior of the transverse and longitudinal correlation lengths of the corresp
onding d + 1-dimensional effective interface model in a non-mean-field regi
me. We also describe the set of pinned sites at small epsilon, for a broad
class of d-dimensional massless models.