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On a continuum percolation model

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of California, Santa Barbara
*
∗Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.
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Abstract

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 536 - 556
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

POISSON PROCESS; CLUSTER DENSITY; LARGE DEVIATIONS AT HIGH DENSITY

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