Localization of a Gaussian membrane model with weak pinning potentials

We consider a class of effective models on ℤ^d called Gaussian membrane models with square-well pinning or σ-pinning. It is known that when d = 1 this model exhibits a localization/delocalization transition that depends on the strength of the pinning. In this paper, we show that when d ≥ 2, once we impose weak pinning potentials the field is always localized in the sense that the corresponding free energy is always positive. We also discuss the case that both square-well potentials and repulsive potentials are acting in high dimensions.