Fast decay of covariances under delta-pinning in the critical and supercritical membrane model
- Bolthausen, Erwin
- Cipriani, Alessandra
- Kurt, Noemi
2010 Mathematics Subject Classification
- 31B30 39A12 60K35 60K37 82B41
- membrane model, pinning, bilaplacian, decay of covariances
We consider the membrane model, that is the centered Gaussian field on 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 epsilon for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions larger than 4 covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in dimensions larger than 5 and logarithmic in 4 dimensions. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.