The critical probability for confetti percolation equals 1/2

Müller, Tobias

doi:10.1002/rsa.20675

Cited by 7 publications

(11 citation statements)

References 10 publications

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“…The connected components of visible portions of leaves (at time 0) form a tessellation of R d , which we call the DLM tessellation. [4,6,15,34], while percolation on the DLM tessellation has been considered in [2,24]. In some of these works the authors call the DLM the 'confetti' model.…”

Section: Overviewmentioning

confidence: 99%

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Leaves on the line and in the plane

Penrose¹

2020

Electron. J. Probab.

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The dead leaves model (DLM) provides a random tessellation of d-space, representing the visible portions of fallen leaves on the ground when d = 2. For d = 1, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For d = 2 we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of dead leaves random measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.

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Section: Overviewmentioning

confidence: 99%

“…Then ξ is Lebesgue measure restricted to those visible leaves which are coloured 1. The CDLM was introduced by Jeulin in [12] (see also [15]), and is the basis of the percolation problems considered in [2,24]. given by the level of greyscale on the leaf visible at x.…”

Section: Dead Leaves Random Measuresmentioning

confidence: 99%

Leaves on the line and in the plane

Penrose¹

2020

Electron. J. Probab.

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“…Then ξ is Lebesgue measure restricted to those visible leaves which are coloured 1. The CDLM was introduced by Jeulin in [10] and is the basis of the percolation problems considered in [2,18]. In the usual version of this, the decisions on how to colour a leaf, and its shape/size, are independent, but they could also be made dependent.…”

Section: Dead Leaves Random Measuresmentioning

confidence: 99%

“…The numbers indicate the reverse order of arrival of the leaves visible within the window. In this paper we view the two visible components of leaf 5 as being separate components of the DLM tessellation Properties of the DLM itself are discussed in [4,6,10,25], while percolation on the DLM tessellation has been considered in [2,18]. In some of these works the authors call the DLM the 'confetti' model.…”

Section: Introduction 1overviewmentioning

confidence: 99%

Leaves on the line and in the plane

Penrose

2018

Preprint

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The dead leaves model (DLM) provides a random tessellation of d-space, representing the visible portions of fallen leaves on the ground when d = 2. For d = 1, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For d = 2 we establish analogous results for the random surface measure of cell boundaries, and also establish the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of dead leaves random measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.

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“…The continuum version of the last model without the hard particles (and therefore with finite range dependences) amounts to the so-called 'confetti percolation' or 'dead leaves' model, for which similar questions have been considered in [8] and [10]. In the latter paper, Müller deploys a different sharp threshold type result with weaker symmetry requirements; it would be very interesting to explore the possible application of those ideas in models such as those mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Percolation of even sites for enhanced random sequential adsorption

Daniels

Penrose

2017

Stochastic Processes and their Applications

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Consider random sequential adsorption on a chequerboard lattice with arrivals at rate 1 on light squares and at rate λ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites black, and the remaining squares white. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability p; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs (λ, p), containing the pair (1, 0.5), such that for (λ, p) lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates.

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The critical probability for confetti percolation equals 1/2

Cited by 7 publications

References 10 publications

Leaves on the line and in the plane

Leaves on the line and in the plane

Leaves on the line and in the plane

Percolation of even sites for enhanced random sequential adsorption

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