Exponential decay ofcovariances for the supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on Zdwhose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose aδ-pinning condition, giving a reward of strengthεfor the field to be 0 at any site of the lattice. In this paper we prove that in dimensionsd≥5covariances 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