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Co-Existence of the occupied and vacant phase in boolean models in three or more dimensions

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Anish Sarkar
Anish Sarkar*
Affiliation:
Indian Statistical Institute
*
*Postal address: Math-Stat Department, Indian Statistical Institute, Calcutta Centre, 203 B.T. Road, Calcutta—700-035, India.
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Abstract

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d ≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

Keywords

CONTINUUM PERCOLATIONBOOLEAN MODEL

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 878 - 889
Copyright
Copyright © Probability Trust 1997 

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