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Continuity and convergence of the percolation function in continuum percolation

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Anish Sarkar
Anish Sarkar*
Affiliation:
University of Utrecht
*
∗ Postal address: Department of Mathematics, University of Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands.
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Abstract

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.

Keywords

POISSON PROCESSPERCOLATION FUNCTIONSCALINGCOUPLING

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 363 - 371
Copyright
Copyright © Applied Probability Trust 1997 

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