Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces

Johan H. Tykesson ()
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Johan H. Tykesson: Chalmers University of Technology and Göteborg University

Journal of Theoretical Probability, 2009, vol. 22, issue 2, 402-417

Abstract: Abstract We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λ u be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ>λ u , then there is a.s. a unique unbounded component at λ. Then we let M=ℍ2×ℝ and show that at λ u there is a.s. not a unique unbounded component. These results are continuum analogs of theorems by Häggström, Peres and Schonmann.

Keywords: Continuum percolation; Poisson Boolean model; Uniqueness in percolation; Mass transport; Homogeneous spaces; 82B21; 82B43 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0179-1

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