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