Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T22:19:06.806Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, K. (1993). Finite clusters in high-density continuous percolation: compression and sphericality. Prob. Theory Rel. Fields 97, 35–63.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I. Wiley, New York.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Meester, R. and Roy, R. (1994). Uniqueness of the unbounded occupied and vacant components in Boolean models. Ann. of Appl. Prob. 4, 933–951.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press, New York.Google Scholar
Penrose, M. D. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. of Appl. Prob. 6, 528–544.Google Scholar
Roy, R. (1990). The RSW theorem and the equality of critical densities and the ‘dual’ critical densities for continuum percolation on R2 . Ann. Prob. 18, 1563–1575.CrossRefGoogle Scholar
Sarkar, A. (1994). Some problems in continuum percolation. , Indian Statistical Institute, New Delhi.Google Scholar