Exponential decay of covariances for the supercritical membrane model

Abstract

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 $\epsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\ge 5$ 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.

23 pages, 2 figures. Comments are welcome

References in corpus (3)

• Localization and delocalization of random interfaces
• Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
• Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension

Cited by in corpus (2)

• Existence of gradient Gibbs measures on regular trees which are not translation invariant
• Pinning for the critical and supercritical membrane model