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Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Part of: Geometric probability and stochastic geometry Special processes

Published online by Cambridge University Press:  01 July 2016

Amites Dasgupta ,
Rahul Roy  and
Anish Sarkar
Amites Dasgupta*
Affiliation:
Indian Statistical Institute
Rahul Roy*
Affiliation:
Indian Statistical Institute
Anish Sarkar*
Affiliation:
Indian Statistical Institute
*
∗ Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India. Email address: [email protected]
∗∗ Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India.
∗∗ Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India.
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Abstract

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Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

Keywords

Boolean modelPoisson point processpercolationcoverage

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 616 - 635
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Supported by a grant from the Department of Science and Technology, Government of India.

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