Noise sensitivity in continuum percolation

Ahlberg, Daniel; Broman, Erik I.; Griffiths, Simon; Morris, Robert

doi:10.1007/s11856-014-1038-y

Cited by 33 publications

(75 citation statements)

References 35 publications

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“…Estimates of this type have previously been obtained in [1,2,4], and the proof presented here will be similar, although different in some details. It will suffice to consider the critical case p = 1 /2 due to monotonicity.…”

Section: Proposition 42supporting

confidence: 66%

“…Proof overview. We will follow the approach developed in [1], and revisited in [4], by which the continuum problem is reduced to its discrete counterpart via a two-stage construction. The central idea is to consider a Poisson point process η k on Ω chosen according to P kn,p for some k ≥ 1, and obtain a configuration η from η k via thinning.…”

Section: Description Of Resultsmentioning

confidence: 99%

“…The study of noise sensitivity has led to a detailed understanding of certain planar percolation models, both discrete: Benjamini, Kalai and Schramm [5], Schramm and Steif [21], Garban, Pete and Schramm [13], and in the continuum: Ahlberg, Broman, Griffiths and Morris [1], and Ahlberg, Griffiths, Morris and Tassion [2]. In this paper we study threshold phenomena and the effect of small perturbations in the context of Poisson Voronoi percolation on R 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Our contributions in this direction are two-fold. First, we describe the discretization method developed in [1], by which we reduce the continuum problem to its discrete counterpart, and emphasize the close relation between threshold phenomena and noise sensitivity of Boolean functions via the study of randomized algorithms. Combining the two techniques we derive quantitative estimates on the width of the threshold window and the rate of decorrelation in (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…We remark that the application of the discretization approach is here somewhat simpler than as originally developed in [1]. Moreover, the techniques we use apply to a range of continuum percolation models such as Poisson Boolean percolation and confetti percolation, as opposed to the approach in [2] that exploits colour-switching tricks.…”

Section: Introductionmentioning

confidence: 99%

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Noise sensitivity and Voronoi percolation

Ahlberg¹,

Baldasso²

2018

Electron. J. Probab.

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In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on R 2 . In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that boxcrossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

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Section: Proposition 42supporting

confidence: 66%

Section: Description Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noise sensitivity and Voronoi percolation

Ahlberg¹,

Baldasso²

2018

Electron. J. Probab.

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Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

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Peccati

Yogeshwaran

2023

Random Struct Algorithms

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Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the k$$ k $$‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite false(2+αfalse)$$ \left(2+\alpha \right) $$‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is 1false/2$$ 1/2 $$ for the planar confetti percolation model with bounded, πfalse/2$$ \pi /2 $$‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

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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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Noise sensitivity in continuum percolation

Cited by 33 publications

References 35 publications

Noise sensitivity and Voronoi percolation

Noise sensitivity and Voronoi percolation

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

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

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