On continuum percolation. (English) Zbl 0588.60096

This paper considerably advances the study of continuum percolation. At the points of a homogeneous Poisson process in k-dimensional space, centre random k-dimensional spheres with independent and identically distributed radii such that the mean sphere content is finite (and positive). It is shown that there exists a positive critical intensity for the mean clump size to be infinite if and only if the random sphere content has finite variance; and that under a strictly weaker condition than finite variance there exists a positive critical intensity for the formation of infinite clumps with positive probability. Thus these two critical intensities need not be the same, in contrast to some much studied examples in lattice percolation.
The paper also studies continuum percolation in the case of general random sets, not just spheres; and gives improved bounds for the critical intensities in the case of unit discs in the plane. Many of the proofs involve multitype branching processes.
Reviewer: C.McDiarmid


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
60J85 Applications of branching processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI