We consider the Poisson Boolean model of continuum percolation. We show that there is a subcritical phase if and only if E(Rd) is finite, where R denotes the radius of the balls around Poisson points and d denotes the dimension. We also give related results concerning the integrability of the diameter of subcritical clusters.
"Subcritical regimes in the Poisson Boolean model of continuum percolation." Ann. Probab. 36 (4) 1209 - 1220, July 2008. https://doi.org/10.1214/07-AOP352