On macroscopic holes in some supercritical strongly dependent percolation models

Sznitman, Alain-Sol

doi:10.1214/18-aop1312

Cited by 23 publications

(31 citation statements)

References 28 publications

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“…One has the following upper bound on the number of elements of U (see (4.10) in [23] and (i)-(iv) in (4.19) of [18]):…”

Section: Entropic Push-down By Disconnectionmentioning

confidence: 99%

“…Letting δ → h and using (4.13), together with ∆ → ∆, we obtain the claim. 3)) in E ≥α leaves in the box a macroscopic volume in its complement when α < h, a situation discussed in [23]. Roughly speaking, a combination of Theorem 4.1 and Theorem 4.2 of this reference imply that if h = h * = h * * holds, conditionally on such a 'macroscopic hole event', the scaled R d -filling of the left-out set by the component of the boundary of S N in E ≥α is close to a translate of the Euclidean ball B in an L 1 -sense (see also Remark 4.3 of the same reference).…”

Section: Pinning Of the Entropic Push-down Under Disconnectionmentioning

confidence: 99%

“…In this article we investigate level-set percolation for the discrete Gaussian free field on Z d , d ≥ 3. The study of percolation phenomena for level-sets of the Gaussian free field was initiated in the eighties (see [6,15,16]) and has attracted considerable interest in recent years (see [8,17,20,22,23]), especially due to the presence of its slowly decaying spatial correlations.…”

Section: Introductionmentioning

confidence: 99%

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Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

Chiarini¹,

Nitzschner

2019

Probab. Theory Relat. Fields

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We investigate level-set percolation of the discrete Gaussian free field on Z d , d ≥ 3, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free 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 derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of A, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of A. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain 'profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of A, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the 'solidification estimates' developed in [18] by A.-S. Sznitman and the second author.

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“…One has the following upper bound on the number of elements of U (see (4.10) in [23] and (i)-(iv) in (4.19) of [18]):…”

Section: Entropic Push-down By Disconnectionmentioning

confidence: 99%

Section: Pinning Of the Entropic Push-down Under Disconnectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

Chiarini¹,

Nitzschner

2019

Probab. Theory Relat. Fields

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“…In particular, precise decay asymptotics for the probability of isolating a macroscopic set from infinity by lower level-sets of the GFF are known in the supercritical regime h < h GFF * , see [9,27,39], which prove a decay of capacity order, and study the GFF conditionally on such a disconnection event. For the membrane model, one may expect an exponential decay of such a disconnection probability at rate proportional to N d−4 for h < h. We also refer to [42] for a similar question studying the emergence of macroscopic holes in the upper level-set of the GFF, and to [24,28,40,41,43,44] for related questions for random interlacements. This article is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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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“…In recent years, large deviation problems for GFF percolation events has attracted considerable attention -see e.g. [19,16,15,22,8]. A common feature in these problems is a deep connection with potential theory and in particular the notion of capacity.…”

Section: Introductionmentioning

confidence: 99%

Analyticity of Gaussian free field percolation observables

Panagiotis,

Severo

2021

Preprint

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We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice Z d , d ≥ 3, are analytic on the whole off-critical regime R \ {h * }. This result concerns in particular the percolation density function θ(h) and the (truncated) susceptibility χ(h).As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all h = h * , which we believe to be a result of independent interest. We also discuss the case of general transient graphs.

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On macroscopic holes in some supercritical strongly dependent percolation models

Cited by 23 publications

References 28 publications

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

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

Analyticity of Gaussian free field percolation observables

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