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Abstract
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The membrane model is a Gaussian interface model with a Hamiltonian involving
second derivatives of the interface height. We consider the model in dimension
under the influence
of
-pinning of
strength .
It is known that this pinning potential manages to localize the interface for any
. We refine this result by
establishing the
-dependence
of the variance and of the exponential decay rate of the covariances for small
(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.
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Keywords
stochastic interface model, membrane model, pinning, decay
of correlations
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Mathematical Subject Classification
Primary: 60G15
Secondary: 31B30, 35B27, 60G60, 60K35, 82B41
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Milestones
Received: 29 July 2020
Revised: 3 May 2021
Accepted: 26 June 2021
Published: 19 February 2022
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