Equality of critical parameters for percolation of Gaussian free field level-sets

Duminil-Copin, Hugo; Goswami, Subhajit; Rodriguez, Pierre-François; Severo, Franco

doi:10.48550/arxiv.2002.07735

Cited by 21 publications

(97 citation statements)

References 61 publications

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“…c) Inequalities that are very similar to the one presented in Theorem 1.1 have been previously established for models such as Random Interlacements [14], Gaussian Free Field [13] and Random Walk Loop Soup [1]. And although such results have proven themselves to be very useful in studying the underlying models [15,5,16,7,6], the techniques developed so far could not be adapted to cylinders' percolation due to the rigidity of cylinder's themselves.…”

Section: Thm:2boxdec_intromentioning

confidence: 98%

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Cylinders' percolation: decoupling and applications

Alves¹,

Caio²,

Teixeira³

et al. 2021

Preprint

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In this paper we establish a strong decoupling inequality for the cylinder's percolation process introduced by Tykesson and Windisch in [18]. This model features a very strong dependency structure, making it difficult to study, and this is why such decoupling inequalities are desirable. It is important to notice that the type of dependencies featured by cylinder's percolation is particularly intricate, given that the cylinders have infinite range (unlike some models like Boolean percolation) while at the same time being rigid bodies (unlike processes such as Random Interlacements). Our work introduces a new notion of fast decoupling, proves that it holds for the model in question and finishes with an application. More precisely, we prove that for a small enough density of cylinders, a random walk on a connected component of the vacant set is transient for all dimensions d ≥ 3.

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Section: Thm:2boxdec_intromentioning

confidence: 98%

“…e) Is it true that the phase transition for percolation on the Poisson Cylinder's model is sharp? This has been established for strongly dependent percolation models such as level sets of the Gaussian Free Field [7] and Random Interlacements [8].…”

Section: Thm:2boxdec_intromentioning

confidence: 99%

Cylinders' percolation: decoupling and applications

Alves¹,

Caio²,

Teixeira³

et al. 2021

Preprint

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“…In the present work, we investigate the percolation phase transition for the level-set of the Gaussian membrane model on Z d , d ≥ 5, which constitutes an example of a percolation model with strong, algebraically decaying correlations. Strongly correlated percolation models of this type have garnered considerable attention recently, with prominent examples being level-sets of the discrete Gaussian free field (GFF) [12,13,16,29,33] or the Ginzburg-Landau interface model [32], the vacant set of random interlacements [30,37,38] or random walk loop soups and their vacant sets [4,7], all in dimensions d ≥ 3. As our main result, we establish that a non-trivial percolation threshold h * (d) also exists for the level-set of the membrane model in d ≥ 5, and this level is positive in high dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…This has been first investigated in the eighties [5,22], and following [33], a very detailed understanding of its geometric properties has emerged during recent years. In particular, [12,13] and the very recent breakthrough [16] show that the level-set of the GFF undergoes a sharp phase transition at a level h GFF and an (almost surely unique) infinite connected component exists in the upper level-set at level h if h < h GFF * (d), and is absent for h > h GFF * (d). In fact, when combined with the results of [29] and [14,31,35], one has that in the entire subcritical regime h > h GFF * (d), the connection probability in the upper level-set admits an exponential decay in d ≥ 4 with a logarithmic correction in d = 3, while in the supercritical regime h < h GFF We now describe the set-up and our results in more detail.…”

Section: Introductionmentioning

confidence: 99%

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

Chiarini¹,

Nitzschner²

2021

Preprint

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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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Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

Chiarini

Nitzschner

2021

Commun. Math. Phys.

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We study level-set percolation for the harmonic crystal on Z d , d ≥ 3, with uniformly elliptic random conductances. We prove that this model undergoes a nontrivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level α disconnects the discrete blow-up of a compact set A ⊆ R d from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity of A, utilizing results from Neukamm, Schäffner and Schlömerkemper (SIAM J Math Anal 49(3):1761-1809, 2017). Furthermore, we give upper bounds on the probability that a local average of the field deviates from some profile function depending on A, when disconnection occurs. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field. The results in this article generalize the findings of Nitzschner (Electron J Probab 23:105, 2018) and Chiarini and Nitzschner (Probab Theory Relat Fields 177(1-2):525-575, 2020) which treat the case of constant conductances. Our proofs involve novel "solidification estimates" for random walks, which are similar in nature to the corresponding estimates for Brownian motion derived by Nitzschner and Sznitman (

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Equality of critical parameters for percolation of Gaussian free field level-sets

Cited by 21 publications

References 61 publications

Cylinders' percolation: decoupling and applications

Cylinders' percolation: decoupling and applications

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

Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

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