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Gilbert's disc model with geostatistical marking

Part of: Special processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  29 November 2018

Daniel Ahlberg  and
Johan Tykesson
Daniel Ahlberg*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada and Uppsala University
Johan Tykesson*
Affiliation:
Chalmers University of Technology and University of Gothenburg
*
* Current address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden.
** Postal address: Department of Mathematics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
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Abstract

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in ℝ2 with radii determined by an underlying stationary and ergodic random field φ:ℝ2→[0,∞), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of ℝ2 does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

Keywords

Continuum percolationcoverage probabilitythreshold comparison

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K37: Processes in random environments
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1075 - 1094
Copyright
Copyright © Applied Probability Trust 2018 

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