Percolation in the vacant set of Poisson cylinders
type: Item
instance of: scholarly article
title: Percolation in the vacant set of Poisson cylinders
zbMATH Open document ID: 1263.82027
author: David Windisch
author: Johan Tykesson
publication date: 2013-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.5338
review text: A Poisson point process is investigated on the space of random lines in \(\mathbb{R}^d\), \(d\geq 3\) where a multiplicative parameter \(u=u(d)\) labels low and high intensity regimes, including a percolation threshold (a critical value \(u_*\) of that parameter). Each line is interpreted as the axis of a bi-infinite cylinder of unit radius. Percolation is studied for the vacant set which is a random set \({\mathcal{V}} \subset \mathbb{R}^d\) not covered by the cylinders. The main question addressed is whether the set \({\mathcal{V}}\) has unbounded connected components , in which case we say that \({\mathcal{V}}\) percolates. Low and high intensity regimes are investigated. In particular, it is proved that in dimensions \(d\geq 3\), \({\mathcal{V}}\) does not percolate for high intensities. It is found that, for \(d\geq 4\), \({\mathcal{V}}\) does percolate at low intensities. Various inspirations follow from the work by \textit{A.-S. Sznitman} [Ann. Math. (2) 171, No. 3, 2039--2087 (2010; Zbl 1202.60160)] on random interlacements, which share common features with Poisson cylinders. Specifically the infinite-range dependence of the current problem requires some techniques, previously developed by Sznitman.
reviewed by: Piotr Garbaczewski
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 82C41
zbMATH DE Number: 6125012
zbMATH Keywords: percolation
zbMATH Keywords: space of random lines
zbMATH Keywords: Poisson cylinders
zbMATH Keywords: vacant sets
zbMATH Keywords: random interlacements
zbMATH Keywords: intensity regimes
zbMATH Keywords: percolation threshold
MaRDI profile type: Publication
OpenAlex ID: W3102519616
arXiv ID: 1010.5338
cites work: Visibility to infinity in the hyperbolic plane, despite obstacles
cites work: Exceptional planes of percolation
cites work: Subcritical regimes in the Poisson Boolean model of continuum percolation
cites work: Percolation
cites work: Continuum Percolation
cites work: Q4110292
cites work: Stochastic and Integral Geometry
cites work: Connectivity bounds for the vacant set of random interlacements
cites work: Percolation for the vacant set of random interlacements
cites work: On the critical parameter of interlacement percolation in high dimension
cites work: Vacant set of random interlacements and percolation
cites work: On the uniqueness of the infinite cluster of the vacant set of random interlacements
DOI: 10.1007/S00440-011-0366-3
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article