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

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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

Permanent link to this document
https://projecteuclid.org/euclid.aap/1316792662

Digital Object Identifier
doi:10.1239/aap/1316792662

Mathematical Reviews number (MathSciNet)
MR2858213

Zentralblatt MATH identifier
1227.60109

Subjects
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]

Keywords
Boolean model Poisson point process percolation coverage

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


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