Noise sensitivity and Voronoi percolation

Ahlberg, Daniel; Baldasso, Rangel

doi:10.1214/18-ejp233

Cited by 16 publications

(33 citation statements)

References 25 publications

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“…In the above argument the OSSS inequality was applied to crossing events of squares, whereas the control of the revealments δ i was achieved by analysing one-arm events; this is roughly how we shall apply the OSSS inequality (see also [AB18] where this argument is carried out for planar Voronoi percolation). In [DCRT19a, DCRT19b, DCRT18] the OSSS inequality is instead applied to one-arm events directly, a powerful approach that ultimately yields the phase transition in all dimensions for a wide class of models.…”

Section: Overview Of Our Methods and Outline Of The Proofmentioning

confidence: 99%

The sharp phase transition for level set percolation of smooth planar Gaussian fields

Muirhead¹,

Vanneuville

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two -roughly equivalent to the integrability of the covariance kernel -whereas previously the phase transition was only known in the case of the Bargmann-Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially.Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.

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Section: Overview Of Our Methods and Outline Of The Proofmentioning

confidence: 99%

The sharp phase transition for level set percolation of smooth planar Gaussian fields

Muirhead¹,

Vanneuville

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…Such bounds were first obtained by Schramm and Steif [14], then by Garban, Pete and Schramm [9] and more recently Tassion and Vanneuville [16]. In another direction, the study of noise sensitivity was extended to percolation models in the continuum by Ahlberg, Broman, Griffiths and Morris [2], Ahlberg and Baldasso [1], and most recently Last, Peccati and Yogeshwaran [11]. One of the conjectures from [5] concerned such a continuum model, known as Voronoi percolation, and that conjecture has motivated the current work.…”

Section: Introductionmentioning

confidence: 94%

On the rate of convergence in quenched Voronoi percolation

Ahlberg¹,

Riva²,

Griffiths³

2021

Preprint

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Position n points uniformly at random in the unit square S, and consider the Voronoi tessellation of S corresponding to the set η of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let HS denote the event that there exists a red horizontal crossing of S in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event HS will occur or not. More precisely, since HS occurs with probability 1/2, by symmetry, they conjectured that the conditional probabilities P(HS|η) converge in probability to 1/2, as n → ∞. This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris and Tassion. In this paper we derive a stronger bound on the rate at which P(HS|η) approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.

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“…To prove (4.3) we apply the OSSS inequality to the random variables (Z i ) i∈Λ defined in (4.1) and the following algorithm (see [MV20], and also [AB18,BKS99,SS10]). First fix a random horizontal line L = {y = U } where U (R) is an independent [0, R]-uniform random variable.…”

Section: Proposition 44 (Russo-type Inequality)mentioning

confidence: 99%

Percolation of the excursion sets of planar symmetric shot noise fields

Muirhead

2022

Stochastic Processes and their Applications

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We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.

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Noise sensitivity and Voronoi percolation

Cited by 16 publications

References 25 publications

The sharp phase transition for level set percolation of smooth planar Gaussian fields

The sharp phase transition for level set percolation of smooth planar Gaussian fields

On the rate of convergence in quenched Voronoi percolation

Percolation of the excursion sets of planar symmetric shot noise fields

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