Computation of Coverage Probabilities in a Spherical Germ-Grain Model

  • Ian Flint
  • Nicolas PrivaultEmail author


We consider a spherical germ-grain model on \(\mathbb {R}^{d}\) in which the centers of the spheres are driven by a possibly non-Poissonian point process. We show that various covering probabilities can be expressed using the cumulative distribution function of the random radii on one hand, and distances to certain subsets of \(\mathbb {R}^{d}\) on the other hand. This result allows us to compute the spherical and linear contact distribution functions, and to derive expressions which are suitable for numerical computation. Determinantal point processes are an important class of examples for which the relevant quantities take the form of Fredholm determinants.


Boolean model Germ-grain model Capacity functional Multipoint probability function Determinantal point process 

Mathematics Subject Classification 2010

60G60 60D05 60G55 


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The authors would like to thank the two anonymous referees for their suggestions which have greatly improved an earlier draft of this paper. This research was supported by Singapore MOE Tier 2 Grant MOE2016-T2-1-036.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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