## Electronic Communications in Probability

### Non-triviality of the vacancy phase transition for the Boolean model

Mathew D. Penrose

#### Abstract

In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 49, 8 pp.

Dates
Accepted: 18 July 2018
First available in Project Euclid: 31 July 2018

https://projecteuclid.org/euclid.ecp/1533002443

Digital Object Identifier
doi:10.1214/18-ECP153

Mathematical Reviews number (MathSciNet)
MR3841410

Zentralblatt MATH identifier
1394.60101

#### Citation

Penrose, Mathew D. Non-triviality of the vacancy phase transition for the Boolean model. Electron. Commun. Probab. 23 (2018), paper no. 49, 8 pp. doi:10.1214/18-ECP153. https://projecteuclid.org/euclid.ecp/1533002443