Advances in Applied Probability

Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Amites Dasgupta, Rahul Roy, and Anish Sarkar

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider the region L = {(x ,y) : 0 ≤ yClog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

Article information

Adv. in Appl. Probab., Volume 43, Number 3 (2011), 616-635.

First available in Project Euclid: 23 September 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Boolean model Poisson point process percolation coverage


Dasgupta, Amites; Roy, Rahul; Sarkar, Anish. Geometry of the Poisson Boolean model on a region of logarithmic width in the plane. Adv. in Appl. Probab. 43 (2011), no. 3, 616--635. doi:10.1239/aap/1316792662.

Export citation


  • Athreya, S., Roy, R. and Sarkar, A. (2004). On the coverage of space by random sets. Adv. Appl. Prob. 36, 1–18.
  • Grimmett, G. R. (1983). Bond percolation on subsets of the square lattice, and the threshold between one-dimensional and two-dimensional behaviour. J. Phys. A. 16, 599–604.
  • Grimmett, G. R. (1999). Percolation, 2nd edn. Springer, Berlin.
  • Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.
  • Molchanov, I. and Scherbakov, V. (2003). Coverage of the whole space. Adv. Appl. Prob. 35, 898–912.
  • Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
  • Petrov, V. V. (2004). A generalization of the Borel–Cantelli lemma. Statist. Prob. Lett. 67, 233–239.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.
  • Tanemura, H. (1993). Behavior of the supercritical phase of a continuum percolation model on $\mathbbR\sp d$. J. Appl. Prob. 30, 382–396.
  • Tanemura, H. (1996). Critical behavior for a continuum percolation model. In Probability Theory and Mathematical Statistics. (Tokyo, 1995), World Scientific, River Edge, NJ, pp. 485–495.