Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T21:30:55.254Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ahlberg, D.,Tassion, V. and Teixeira, A. (2018).Existence of an unbounded vacant set for subcritical continuum percolation.Electron. Commun. Prob. 23, 8pp.Google Scholar
[2]Ahlberg, D.,Tassion, V. and Teixeira, A. (2018).Sharpness of the phase transition for continuum percolation in ℝ2.Prob. Theory Relat. Fields 172,525–581.Google Scholar
[3]Aizenman, M. et al. (1983).On a sharp transition from area law to perimeter law in a system of random surfaces.Commun. Math. Phys. 92,19–69.Google Scholar
[4]Benjamini, I. and Stauffer, A. (2013).Perturbing the hexagonal circle packing: a percolation perspective.Ann. Inst. H. Poincaré Prob. Statist. 49,1141–1157.Google Scholar
[5]Benjamini, I.,Jonasson, J.,Schramm, O. and Tykesson, J. (2009).Visibility to infinity in the hyperbolic plane, despite obstacles.ALEA 6,323–342.Google Scholar
[6]Błaszczyszyn, B. and Yogeshwaran, D. (2013).Clustering and percolation of point processes.Electron. J. Prob. 18, 20pp.Google Scholar
[7]Błaszczyszyn, B. and Yogeshwaran, D. (2015).Clustering comparison of point processes, with applications to random geometric models. In Stochastic Geometry, Spatial Statistics and Random Fields (Lecture Notes Math. 2120),Springer,Cham, pp. 31–71.Google Scholar
[8]Bollobás, B. and Riordan, O. (2006).Percolation.Cambridge University Press,New York.Google Scholar
[9]Broman, E. I. and Tykesson, J. (2016).Connectedness of Poisson cylinders in Euclidean space.Ann. Inst. H. Poincaré Prob. Statist. 52,102–126.Google Scholar
[10]Chiu, S. N.,Stoyan, D.,Kendall, W. S. and Mecke, J. (2013).Stochastic geometry and its applications,3rd edn.John Wiley,Chichester.Google Scholar
[11]Gilbert, E. N. (1961).Random plane networks.J. Soc. Indust. Appl. Math. 9,533–543.Google Scholar
[12]Gilbert, E. N. (1965).The probability of covering a sphere with N circular caps.Biometrika 52,323–330.Google Scholar
[13]Gouéré, J.-B. (2008).Subcritical regimes in the Poisson Boolean model of continuum percolation.Ann. Prob. 36,1209–1220.Google Scholar
[14]Häggström, O. and Jonasson, J. (2006).Uniqueness and non-uniqueness in percolation theory.Prob. Surveys 3,289–344.Google Scholar
[15]Hall, P. (1985).On continuum percolation.Ann. Prob. 13,1250–1266.Google Scholar
[16]Hilário, M. R.,Sidoravicius, V. and Teixeira, A. (2015).Cylinders' percolation in three dimensions.Prob. Theory Relat. Fields 163,613–642.Google Scholar
[17]Illian, J.,Penttinen, A.,Stoyan, H. and Stoyan, D. (2008).Statistical Analysis and Modelling of Spatial Point Patterns.John Wiley,Chichester.Google Scholar
[18]Kesten, H. (1982).Percolation Theory for Mathematicians (Progress Prob. Statist. 2).Birkhäuser,Boston, MA.Google Scholar
[19]Meester, R. and Roy, R. (1996).Continuum Percolation (Camb. Tracts Math. 119).Cambridge University Press.Google Scholar
[20]Penrose, M. (2003).Random Geometric Graphs (Oxford Stud. Prob. 5).Oxford University Press.Google Scholar
[21]Roy, R. and Tanemura, H. (2002).Critical intensities of Boolean models with different underlying convex shapes.Adv. Appl. Prob. 34,48–57.Google Scholar
[22]Russo, L. (1981).On the critical percolation probabilities.Z. Wahrscheinlichkeitsth. 56,229–237.Google Scholar
[23]Schlather, M.,RibeiroP. J., Jr. P. J., Jr. and Diggle, P. J. (2004).Detecting dependence between marks and locations of marked point processes.J. R. Statist. Soc. B 66,79–93.Google Scholar
[24]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar
[25]Tykesson, J. and Windisch, D. (2012).Percolation in the vacant set of Poisson cylinders.Prob. Theory Relat. Fields 154,165–191.Google Scholar
[26]Van den Berg, J.,Peres, Y.,Sidoravicius, V. and Vares, M. E. (2008).Random spatial growth with paralyzing obstacles.Ann. Inst. H. Poincaré Prob. Statist. 44,1173–1187.Google Scholar