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Computation of Coverage Probabilities in a Spherical Germ-Grain Model

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Abstract

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.

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Acknowledgments

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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  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Ian Flint & Nicolas Privault

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  1. Ian Flint
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  2. Nicolas Privault
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Correspondence to Nicolas Privault.

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Flint, I., Privault, N. Computation of Coverage Probabilities in a Spherical Germ-Grain Model. Methodol Comput Appl Probab 23, 491–502 (2021). https://doi.org/10.1007/s11009-019-09741-5

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  • DOI: https://doi.org/10.1007/s11009-019-09741-5

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