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

#### Abstract

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

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

Dates
First available in Project Euclid: 23 September 2011

https://projecteuclid.org/euclid.aap/1316792662

Digital Object Identifier
doi:10.1239/aap/1316792662

Mathematical Reviews number (MathSciNet)
MR2858213

Zentralblatt MATH identifier
1227.60109

#### Citation

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. https://projecteuclid.org/euclid.aap/1316792662

#### References

• 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.