Vol. 2, No. 4, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN (electronic): 2690-1005
ISSN (print): 2690-0998
Author Index
To Appear
Other MSP Journals
Pinning for the critical and supercritical membrane model

Florian Schweiger

Vol. 2 (2021), No. 4, 745–820

The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension d 4 under the influence of δ-pinning of strength 𝜀. It is known that this pinning potential manages to localize the interface for any 𝜀 > 0. We refine this result by establishing the 𝜀-dependence of the variance and of the exponential decay rate of the covariances for small 𝜀 (similar to the corresponding results for the discrete Gaussian free field by Bolthausen and Velenik). We also show the existence of a thermodynamic limit of the field. These conclusions improve upon earlier works by Bolthausen, Cipriani and 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 multiscale argument to construct suitable test functions.

stochastic interface model, membrane model, pinning, decay of correlations
Mathematical Subject Classification
Primary: 60G15
Secondary: 31B30, 35B27, 60G60, 60K35, 82B41
Received: 29 July 2020
Revised: 3 May 2021
Accepted: 26 June 2021
Published: 19 February 2022
Florian Schweiger
Institut für angewandte Mathematik
Universität Bonn
Department of Mathematics
Weizmann Institute of Science