  We consider the membrane model, that is the centered Gaussian field on
$\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete
Bilaplacian. We impose a $\delta-$pinning condition, giving a reward of
strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In
this paper we prove that in dimensions $d\geq 5$ covariances of the pinned
field decay at least exponentially, as opposed to the field without pinning,
where the decay is polynomial. The proof is based on estimates for certain
discrete weighted norms, a percolation argument and on a Bernoulli domination
result.
