Skip to main content

Gilbert's disc model with geostatistical marking

  • Daniel Ahlberg (a1) and Johan Tykesson (a2)

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.

Corresponding author
* 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.
Hide All
[1]Ahlberg, D.,Tassion, V. and Teixeira, A. (2018).Existence of an unbounded vacant set for subcritical continuum percolation.Electron. Commun. Prob. 23, 8pp.
[2]Ahlberg, D.,Tassion, V. and Teixeira, A. (2018).Sharpness of the phase transition for continuum percolation in ℝ2.Prob. Theory Relat. Fields 172,525581.
[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. 92,1969.
[4]Benjamini, I. and Stauffer, A. (2013).Perturbing the hexagonal circle packing: a percolation perspective.Ann. Inst. H. Poincaré Prob. Statist. 49,11411157.
[5]Benjamini, I.,Jonasson, J.,Schramm, O. and Tykesson, J. (2009).Visibility to infinity in the hyperbolic plane, despite obstacles.ALEA 6,323342.
[6]Błaszczyszyn, B. and Yogeshwaran, D. (2013).Clustering and percolation of point processes.Electron. J. Prob. 18, 20pp.
[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. 3171.
[8]Bollobás, B. and Riordan, O. (2006).Percolation.Cambridge University Press,New York.
[9]Broman, E. I. and Tykesson, J. (2016).Connectedness of Poisson cylinders in Euclidean space.Ann. Inst. H. Poincaré Prob. Statist. 52,102126.
[10]Chiu, S. N.,Stoyan, D.,Kendall, W. S. and Mecke, J. (2013).Stochastic geometry and its applications,3rd edn.John Wiley,Chichester.
[11]Gilbert, E. N. (1961).Random plane networks.J. Soc. Indust. Appl. Math. 9,533543.
[12]Gilbert, E. N. (1965).The probability of covering a sphere with N circular caps.Biometrika 52,323330.
[13]Gouéré, J.-B. (2008).Subcritical regimes in the Poisson Boolean model of continuum percolation.Ann. Prob. 36,12091220.
[14]Häggström, O. and Jonasson, J. (2006).Uniqueness and non-uniqueness in percolation theory.Prob. Surveys 3,289344.
[15]Hall, P. (1985).On continuum percolation.Ann. Prob. 13,12501266.
[16]Hilário, M. R.,Sidoravicius, V. and Teixeira, A. (2015).Cylinders' percolation in three dimensions.Prob. Theory Relat. Fields 163,613642.
[17]Illian, J.,Penttinen, A.,Stoyan, H. and Stoyan, D. (2008).Statistical Analysis and Modelling of Spatial Point Patterns.John Wiley,Chichester.
[18]Kesten, H. (1982).Percolation Theory for Mathematicians (Progress Prob. Statist. 2).Birkhäuser,Boston, MA.
[19]Meester, R. and Roy, R. (1996).Continuum Percolation (Camb. Tracts Math. 119).Cambridge University Press.
[20]Penrose, M. (2003).Random Geometric Graphs (Oxford Stud. Prob. 5).Oxford University Press.
[21]Roy, R. and Tanemura, H. (2002).Critical intensities of Boolean models with different underlying convex shapes.Adv. Appl. Prob. 34,4857.
[22]Russo, L. (1981).On the critical percolation probabilities.Z. Wahrscheinlichkeitsth. 56,229237.
[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 66,7993.
[24]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.
[25]Tykesson, J. and Windisch, D. (2012).Percolation in the vacant set of Poisson cylinders.Prob. Theory Relat. Fields 154,165191.
[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. 44,11731187.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed