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.
"Non-triviality of the vacancy phase transition for the Boolean model." Electron. Commun. Probab. 23 1 - 8, 2018. https://doi.org/10.1214/18-ECP153