Quenched Voronoi percolation

Ahlberg, Daniel; Griffiths, Simon; Morris, Robert; Tassion, Vincent

doi:10.1016/j.aim.2015.09.005

Cited by 25 publications

(83 citation statements)

References 23 publications

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“…• In [AGMT16], the authors study quenched Voronoi percolation and answer a conjecture from [BKS99] related to the notion of noise sensitivity: they prove that, asymptotically almost surely, the quenched probabilities of crossing event do not depend on η. They also prove quenched and annealed noise sensitivity results for frozen dynamical Voronoi percolation, see Theorem 1.4 below.…”

Section: Previous Results On Noise Sensitivity For Voronoi Percolationmentioning

confidence: 99%

“…Quenched estimates on arm events. In [Van18] we have studied quenched arm events (by following a strategy from [AGMT16]) and we have roughly proved that with high probability the quenched probabilities do not depend on the environment η (up to a constant). In particular, we have proved the following result, where we use the notation…”

Section: Estimates For Crossing and Arm Eventsmentioning

confidence: 99%

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Annealed scaling relations for Voronoi percolation

Vanneuville

2019

Electron. J. Probab.

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In this paper, we study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in R 2 whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in [GPS10] by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object.Our main motivation is the study of two models of dynamical critical Voronoi percolation in the plane. In the first model, the Voronoi tiling does not evolve in time while the colors of the cells are resampled at rate 1. In the second model, the centers of the cells move according to (independent) long range stable Lévy processes but the colors do not evolve in time. We prove that for these two dynamical processes there exist almost surely exceptional times with an unbounded monochromatic component.

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Section: Previous Results On Noise Sensitivity For Voronoi Percolationmentioning

confidence: 99%

Section: Estimates For Crossing and Arm Eventsmentioning

confidence: 99%

Annealed scaling relations for Voronoi percolation

Vanneuville

2019

Electron. J. Probab.

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“…The study of random Voronoi tessellations goes back several decades in time, yet it was only about a decade ago that Bollobàs and Riordan [BR06a] gave the first proof for the fact that the critical probability in Poisson Voronoi percolation in the plane equals 1/2; see also [AGMT16]. One of the main difficulties faced in studying the phase transition is to derive RSW techniques that apply in this setting.…”

Section: Poisson Voronoi Percolationmentioning

confidence: 99%

Sharpness of the phase transition for continuum percolation in $$\mathbb {R}^2$$ R 2

Ahlberg

Tassion

Teixeira

2017

Probab. Theory Relat. Fields

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We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter λ c . Moreover,• For λ < λ c , the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.• At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.• For λ > λ c , the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.Mathematics Subject Classification (2010): 60K35, 82B43, 60G55

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“…Finally, in Chapter 7, we shall present a new result (Theorem 7.1) which improves on the probability bound given by Theorem 1.1 of [1] (which appears here as Theorem 6.2). We remark that many of the ideas involved in the proof arose in discussions with Daniel Ahlberg and Simon Griffiths.…”

Section: Introductionmentioning

confidence: 92%

“…We then turn to Quenched Voronoi Percolation in Chapter 6. We will discuss results from the paper "Quenched Voronoi Percolation", by Ahlberg, Griffiths, Morris, and Tassion [1] in 2015. At some moments in that paper there are important details which are left a little vague.…”

Section: Introductionmentioning

confidence: 99%

An Invitation to Noise Sensitivity and Applications to Quenched Voronoi Percolation

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Quenched Voronoi percolation

Cited by 25 publications

References 23 publications

Annealed scaling relations for Voronoi percolation

Annealed scaling relations for Voronoi percolation

Sharpness of the phase transition for continuum percolation in $$\mathbb {R}^2$$ R 2

An Invitation to Noise Sensitivity and Applications to Quenched Voronoi Percolation

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