  We consider the harmonic crystal on the d-dimensional lattice, d larger or
equal to 3, that is the centered Gaussian field $\phi$ with covariance given by
the Green function of the simple random walk on $Z^d$. Our main aim is to
obtain quantitative information on the repulsion phenomenon that arises when we
condition the field to be larger than an IID field $\sigma$ (which is also
independent of $\phi$), for every x in a large region $D_N=ND\cap \Z^d$, with N
a positive integer and $D \subset\R^d$. We are mostly motivated by results for
given typical realizations of the $\sigma$ (quenched set-up), since the
conditioned harmonic crystal may be seen as a model for an equilibrium
interface, constrained not to go below a inhomogeneous substrate that acts as a
hard wall. We consider various types of substrate and we observe that the
interface is pushed away from the wall much more than in the case of a flat
wall as soon as the upward tail of $\sigma$ is heavier than Gaussian, while
essentially no effect is observed if the tail is sub--Gaussian. In the critical
case, that is the one of approximately Gaussian tail, the interplay of the two
sources of randomness, $\phi$ and $\sigma$, leads to an enhanced repulsion
effect of additive type.
