Probability to be positive for the membrane model in dimensions 2 and 3

Buchholz, Simon; Deuschel, Jean–Dominique; Kurt, Noemi; Schweiger, Florian

doi:10.1214/19-ecp245

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…Nonetheless, in recent years several results that were already known for the gradient model could be established also for the membrane model. Let us mention the scaling limit of the membrane model [CDH19], the maximum of the field [CCH16, CDH19,Sch20], and its behaviour under entropic repulsion [Sak03,Kur07,Kur09,BDKS19].…”

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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Section: Setting and Overviewmentioning

confidence: 99%

Pinning for the critical and supercritical membrane model

Schweiger

2020

Preprint

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“…Besides for scaling limits, other questions of interest for the membrane model (and many other interface models) include entropic repulsion, pinning, wetting, and the behavior of the interface maximum. Entropic repulsion was addressed in d ≥ 5 by [11] and [17], in d = 4 by [13] (and the thesis [12]), and in d = 2, 3 by [4]. For pinning in d ≥ 4, some recent results are given in [18], and pinning in d = 2, 3 is not well understood.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

Thoma¹

2021

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We characterize the behavior of a random discrete interface φ on [−L, L] d ∩ Z d with energyV (∆φ(x)) as L → ∞, where ∆ is the discrete Laplacian and V is a uniformly convex, symmetric, and smooth potential. The interface φ is called the non-Gaussian membrane model. By analyzing the Helffer-Sjöstrand representation associated to ∆φ, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions d = 2, 3, Gaussian approximation in negative regularity spaces for all d ≥ 2, and the infinite volume limit in d ≥ 5. Our results generalize some of those of [7].

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“…While the membrane model retains some crucial properties of the GFF − in particular one still has a domain Markov property − it lacks some key features which have made the mathematical investigation of the GFF tractable, such as an elementary finite-volume random walk representation or a finite-volume FKG inequality. A number of classical results for the GFF have been verified in the context of the membrane model, in particular the behavior of its maximum and entropic repulsion by a hard wall (see [6,8,11,19,20,21,34,36]), whereas questions concerning its level-set percolation have remained open. In the present article, we make progress in this direction by establishing that a phase transition occurs at a finite level h * (d) in d ≥ 5, and by characterizing parts of its subcritical and supercritical regimes, similar in spirit to the above mentioned program for the GFF.…”

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Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$

Chiarini¹,

Nitzschner²

2021

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We consider level-set percolation for the Gaussian membrane model on Z d , with d ≥ 5, and establish that as h ∈ R varies, a non-trivial percolation phase transition for the level-set above level h occurs at some finite critical level h * , which we show to be positive in high dimensions. Along h * , two further natural critical levels h * * and h are introduced, and we establish that −∞ < h ≤ h * ≤ h * * < ∞, in all dimensions. For h > h * * , we find that the connectivity function of the level-set above h admits stretched exponential decay, whereas for h < h, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, Ráth and Sapozhnikov [14] for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model. * (d) ∈ (0, ∞), * (d), the unique infinite cluster is "well-behaved" in the sense that its chemical distances are close to the Euclidean distance, and the simple random walk on it fulfills a quenched invariance principle, with Gaussian heat kernel bounds. We refer to [17] for an even more precise investigation of connection probabilities in the entire off-critical regime h = h GFF * (d) for the level-set percolation of the GFF.The membrane model on Z d , d ≥ 5, may be seen as a variant of the GFF, in which the gradient structure in the corresponding Gibbs measure is replaced by the discrete Laplacian. This choice gives rise to a discrete interface that favors constant curvature, and models of this type are used in the physics literature to characterize thermal fluctuations of biomembranes formed by lipid bilayers (see, e.g. [23,25]). While the membrane model retains some crucial properties of the GFF − in particular one still has a domain Markov property − it lacks some key features which have made the mathematical investigation of the GFF tractable, such as an elementary finite-volume random walk representation or a finite-volume FKG inequality. A number of classical results for the GFF have been verified in the context of the membrane model, in particular the behavior of its maximum and entropic repulsion by a hard wall (see [6,8,11,19,20,21,34,36]), whereas questions concerning its level-set percolation have remained open. In the present article, we make progress in this direction by establishing that a phase transition occurs at a finite level h * (d) in d ≥ 5, and by characterizing parts of its subcritical and supercritical regimes, similar in spirit to the above mentioned program for the GFF. A key tool in our proofs is a decoupling inequality for the membrane model, which we derive in Section 3 akin to what was done for the GFF in [29]. This decoupling inequality is instrumental to prove the finiteness of two further critical parameters h(d)(≤ h * (d) ≤)h * * (d), characterizing a strongly percolative regime h < h(d) and a strongly non-percolative regime h > h * * (d), respectively.

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Probability to be positive for the membrane model in dimensions 2 and 3

Cited by 6 publications

References 26 publications

Pinning for the critical and supercritical membrane model

Pinning for the critical and supercritical membrane model

Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$

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