Fast decay of covariances under δ−pinning in the critical and supercritical membrane model

Abstract

We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δ−pinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥4 covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in d≥5 and logarithmic in d=4. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.