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Coverage of space in Boolean models. (English) Zbl 1123.60007

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 119-127 (2006).
Summary: For a marked point process \(\{(x_i,S_i)_{i\geq 1}\}\) with \(\{x_i\in \Lambda:i\geq 1\}\) being a point process on \(\Lambda \subseteq \mathbb R^d\) and \(\{S_i\subseteq \mathbb R^d:i\geq 1\}\) being random sets consider the region \(C=\bigcup_{i\geq 1}(x_i+S_i)\). This is the covered region obtained from the Boolean model \(\{(x_i+S_i):i\geq 1\}\). The Boolean model is said to be completely covered if \(\Lambda \subseteq C\) almost surely. If \(\Lambda\) is an infinite set such that \({\mathbf s}+\Lambda \subseteq \Lambda\) for all \({\mathbf s}\in \Lambda\) (e.g., the orthant), then the Boolean model is said to be eventually covered if \({\mathbf t}+\Lambda \subseteq C\) for some \({\mathbf t}\) almost surely. We discuss the issues of coverage when \(\Lambda\) is \(\mathbb R^d\) and when \(\Lambda\) is \([0,\infty)^d\).
For the entire collection see [Zbl 1113.60008].

MSC:

60D05 Geometric probability and stochastic geometry
05C80 Random graphs (graph-theoretic aspects)
05C40 Connectivity
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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