Exponential decay of covariances for the supercritical membrane model

TL;DR

This paper proves that in dimensions five and higher, the covariances of a pinned membrane model decay exponentially, contrasting with polynomial decay in the unpinned case, using advanced probabilistic and analytical techniques.

Contribution

It establishes exponential decay of covariances for the supercritical membrane model with pinning in high dimensions, a significant improvement over previous polynomial decay results.

Findings

Covariances decay exponentially in dimensions d ≥ 5

Pinning induces a phase transition from polynomial to exponential decay

Uses percolation and Bernoulli domination techniques

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