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Critical intensities of Boolean models with different underlying convex shapes

Part of: Special processes

Published online by Cambridge University Press:  19 February 2016

Rahul Roy  and
Hideki Tanemura
Rahul Roy*
Affiliation:
Indian Statistical Institute, New Delhi
Hideki Tanemura*
Affiliation:
Chiba University
*
∗ Postal address: Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.
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Abstract

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Keywords

Poisson processBoolean modelpercolationcritical intensity

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 48 - 57
Copyright
Copyright © Applied Probability Trust 2002 

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