Gilbert's disc model with geostatistical marking

• Published in: Advances in applied probability Vol. 50; no. 4; pp. 1075 - 1094
• Main Authors: Ahlberg, Daniel, Tykesson, Johan
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2018; Applied Probability Trust
• Subjects: 2013; Continuum percolation; coverage probability; cylinders; Dependence; Fields (mathematics); Geostatistics; IU S. N; Marking; Mathematics; Original Article; percolation; Percolation theory; plane; Poisson density functions; Probability; Probability Theory and Statistics; Sannolikhetsteori och statistik; Simulation; Stochastic geometry and its applications; Stochastic processes; threshold comparison; Primary 60D05; coverage probability; 60K37; Continuum percolation; 60K35; threshold comparison
• ISSN: 0001-8678; 1475-6064; 1475-6064
• DOI: 10.1017/apr.2018.51

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.