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Finite clusters in high density Boolean models with balls of varying sizes

Part of: Special processes

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, Delhi Centre, 7 S.J.S. Sansanwal Marg., N. Delhi-110016, India. Email address: [email protected]
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Abstract

In this paper we study finite clusters in a high density Boolean model with balls of two distinct sizes. Alexander (1993) studied the geometric structures of finite clusters in a high density Boolean model with balls of fixed size and showed that the only possible structure admitted by such events is that all Poisson points comprising the cluster are packed tightly inside a small sphere. When the balls are of varying sizes, the event that the cluster consists of k1 big balls and k2 small balls (both k1, k2 ≥ 1) occurs only when the centres of all big balls are compressed in a small sphere and the centres of the small balls are distributed uniformly inside the region formed by the big balls in such a way that the small balls are totally contained inside the big balls. We also show that it is most likely that a finite cluster in a high density Boolean model with varying ball sizes is made up only of small balls.

Keywords

Poisson processfinite clusters

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 929 - 947
Copyright
Copyright © Applied Probability Trust 1998 

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References

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