Fast decay of covariances under $\delta-$pinning in the critical and supercritical membrane model

TL;DR

This paper demonstrates that in four or more dimensions, the covariances of a membrane model with delta-pinning decay at least stretched-exponentially, contrasting with the polynomial or logarithmic decay in unpinned cases.

Contribution

It establishes the decay rate of covariances in the membrane model with delta-pinning for dimensions four and higher, using novel Sobolev norm estimates and Bernoulli domination.

Findings

Covariances decay at least stretched-exponentially in dimensions d≥4.

Unpinned membrane model exhibits polynomial or logarithmic decay depending on dimension.

The proof employs discrete Sobolev norm estimates 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 $d\geq 4$ 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.