## Advances in Applied Probability

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

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

Amites Dasgupta, Rahul Roy, and Anish Sarkar

#### Abstract

Consider the region
*L* = {(*x* ,*y*) : 0 ≤ *y* ≤ *C*log(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
**R**^{2}. 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 λ_{c} 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