On the cost of the bubble set for random interlacements

Sznitman, Alain-Sol

doi:10.48550/arxiv.2105.12110

Cited by 3 publications

(3 citation statements)

References 31 publications

(147 reference statements)

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“…Typically, the exponential rate of decay is given by the solution of a constrained optimization problem involving the Dirichlet energy and, in some cases, the percolation density θ as well-see e.g. [Szn19a,Szn20,Szn21] for results in that direction for the closely related model of random interlacements. It is therefore relevant to understand the regularity of θ in order to study these optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Analyticity of Gaussian Free Field Percolation Observables

Panagiotis

Severo

2022

Commun. Math. Phys.

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We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $${{\mathbb {Z}}}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , are analytic on the whole off-critical regime $${{\mathbb {R}}}\setminus \{h_*\}$$ R \ { h ∗ } . This result concerns in particular the percolation density function $$\theta (h)$$ θ ( h ) and the (truncated) susceptibility $$\chi (h)$$ 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 $$h\ne h_*$$ 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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Section: Introductionmentioning

confidence: 99%

Analyticity of Gaussian Free Field Percolation Observables

Panagiotis

Severo

2022

Commun. Math. Phys.

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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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“…A common feature in these problems is a deep connection with potential theory and in particular the notion of capacity. Typically, the exponential rate of decay is given by the solution of a constrained optimization problem involving the Dirichlet energy and, in some cases, the percolation density θ as well [21,24,25]. It is therefore relevant to understand the regularity of θ in order to study these optimization problems.…”

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.

show abstract

On the cost of the bubble set for random interlacements

Cited by 3 publications

References 31 publications

Analyticity of Gaussian Free Field Percolation Observables

Analyticity of Gaussian Free Field Percolation Observables

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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