Fast decay of covariances under \delta $pinning in the critical and supercritical membrane model
We consider the membrane model, that is the centered Gaussian field on \mathbb Z^d$ 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 $d\geq 4$ covariances of the pinned field decay at least stretched exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\geq 5$ and logarithmic in $d=4. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.