Exponential Decay of Covariances for the Supercritical Membrane Model

Bolthausen, Erwin; Cipriani, Alessandra; Kurt, Noemi

doi:10.1007/s00220-017-2886-x

Cited by 13 publications

(18 citation statements)

References 20 publications

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“…(2) If one considers the pinned versions of the purely gradient and purely Laplacian model, it is known in different settings that the field exhibits exponential decay of correlations (Bolthausen and Brydges, 2001, Bolthausen et al, 2017, Ioffe and Velenik, 2000. Can one say the same for the mixed model?…”

Section: 4mentioning

confidence: 99%

The scaling limit of the membrane model

Cipriani¹,

Dan²,

Hazra³

2019

Ann. Probab.

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In this article we study the scaling limit of the interface model on Z d where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.2000 Mathematics Subject Classification. 31B30, 60J45, 60G15, 82C20. Key words and phrases. Mixed model, Gaussian free field, membrane model, random interface, scaling limit.RSH acknowledges MATRICS grant from SERB and the Dutch stochastics cluster STAR (Stochastics Theoretical and Applied Research) for an invitation to TU Delft where part of this work was carried out. The authors thank Francesco Caravenna for helpful discussions.

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Section: 4mentioning

confidence: 99%

The scaling limit of the membrane model

Cipriani¹,

Dan²,

Hazra³

2019

Ann. Probab.

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“…We also prove lower bounds on the dependence of the mass on ε that we believe to be optimal. This is not the first result in that direction: in [BCK16] Bolthausen, Cipriani and Kurt proved stretched-exponential decay of the covariance in d ≥ 4, and in [BCK17] they improved this to exponential decay, if d ≥ 5. Their rate of decay is far from optimal, though.…”

Section: Setting and Overviewmentioning

confidence: 90%

“…For d = 2, 3, however, there are no known results. In fact, in [BCK17] the authors wrote "it is well possible that exponential decay of correlations is true also for lower dimensions d = 2, 3, but we do not know of a method which could successfully be applied", and we have nothing to add to this statement.…”

Section: Setting and Overviewmentioning

confidence: 99%

Pinning for the critical and supercritical membrane model

Schweiger

2020

Preprint

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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 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-Velenik). We also show the existence of a thermodynamic limit of the field.Our results are significant improvements of results by Bolthausen-Cipriani-Kurt and by Sakagawa. 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 construction to construct suitable test functions.

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“…For more on gradient models on the lattice with homogeneous interactions we refer to [10], [13], [26], [2] and [4]. For more on gradient models on the lattice in random environments, see [5], [15], [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Existence of gradient Gibbs measures on regular trees which are not translation invariant

Henning¹,

Kuelske²

2021

Preprint

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We provide an existence theory for gradient Gibbs measures for Zvalued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

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Exponential Decay of Covariances for the Supercritical Membrane Model

Cited by 13 publications

References 20 publications

The scaling limit of the membrane model

The scaling limit of the membrane model

Pinning for the critical and supercritical membrane model

Existence of gradient Gibbs measures on regular trees which are not translation invariant

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