On the repulsion of an interface above a correlated substrate

We analyze a model of an interface fluctuating above a rough substrate. It is based on harmonic crystals, or lattice free fields, indexed by Z(d), d greater than or equal to 3. The phenomenon for which we want to get precise quantitative estimates is the repulsion effect of the substrate on the interface: the substrate is itself a random field, but its randomness is quenched (this generalizes the widely considered case of a flat deterministic substrate). With respect to [2] in which the substrate has been taken to be an IID field, here the substrate is an harmonic crystal, as the interface, and as such it is strongly correlated. We obtain the leading asymptotic behavior of the model in the limit of a very extended substrate: we show in particular that, to leading order, the effect of an IID substrate cannot be distinguished from the effect of an harmonic crystal substrate. We observe however that, unlike in the IID substrate case, annealed and quenched models display sharply different features.