Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces |
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Authors: | Johan H. Tykesson |
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Affiliation: | 1. Department of Mathematics, Division of Mathematical Statistics, Chalmers University of Technology and G?teborg University, 412 96, G?teborg, Sweden
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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=?2×? 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. |
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Keywords: | Continuum percolation Poisson Boolean model Uniqueness in percolation Mass transport Homogeneous spaces |
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