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.
Highlights
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 ≥ 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
The Boolean model [6, 8] is a classic model of continuum percolation [11, 3] and more general stochastic geometry [9, 4, 14, 10]
Summary
The Boolean model [6, 8] is a classic model of continuum percolation [11, 3] and more general stochastic geometry [9, 4, 14, 10]. One may define a critical value λc of λ, depending on the radius distribution, above which the occupied region percolates, and a further critical value λ∗c , below which the complementary vacant region percolates It is a fundamental question whether these critical values are non-trivial, i.e. strictly positive and finite. The occupied and vacant regions of the (Poisson, spherical) Boolean model are random sets Zλ ⊂ Rd and Zλ∗ ⊂ Rd, given respectively by. Theorem 2 says that for the spherical Poisson Boolean model with E [ρd] < ∞, there exists a non-zero value of the intensity λ for which the vacant region percolates. In parallel and independent work, Ahlberg, Tassion and Teixeira [2] prove a similar set of results to our Theorems 2 and 3; their proof seems to be completely different from ours. We let o denote the origin in Rd, and for r > 0 put B(r) := B(o, r)
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