Pinning for the critical and supercritical membrane model

TL;DR

This paper analyzes the membrane model in dimensions four and higher under delta-pinning, refining understanding of variance and covariance decay, and establishing the existence of a thermodynamic limit, with new probabilistic and analytical techniques.

Contribution

It provides detailed $ ext{ε}$-dependence results for variance and covariances, introduces a correlation inequality and a Widman hole filler argument, and proves the thermodynamic limit for the pinned membrane model.

Findings

Variance and covariance decay rates are explicitly characterized for small $ ext{ε}$.

Existence of a thermodynamic limit of the field is established.

New probabilistic and analytical techniques are developed for the model.

Abstract

The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension $\mathsf{d}\ge4$ under the influence of $\delta$-pinning of strength $\varepsilon$. It is known that this pinning potential manages to localize the interface for any $\varepsilon>0$. We refine this result by establishing the $\varepsilon$-dependence of the variance and of the exponential decay rate of the covariances for small $\varepsilon$ (similar to the corresponding results for the discrete Gaussian free field by Bolthausen-Velenik). We also show the existence of a thermodynamic limit of the field. These conclusions improve upon earlier works by Bolthausen-Cipriani-Kurt and by Sakagawa. The problem has similarities to the homogenization of elliptic operators in randomly perforated domains, and our proof takes inspiration from this connection. The main new ideas are a correlation inequality for the set of pinned points, and a probabilistic Widman hole filler argument which relies on a discrete multipolar Hardy-Rellich inequality and on a multi-scale argument to construct suitable test functions.