Non-triviality of the vacancy phase transition for the Boolean model. (English) Zbl 1394.60101

Penrose, Mathew D.

Electron. Commun. Probab. 23, Paper No. 49, 8 p. (2018).

Summary: 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.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
82B43	Percolation

Keywords:

percolation; Poisson process; vacant region; critical value