Clustering and percolation of point processes

Abstract

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are k-coverage in the Boolean model (coverage by at least k grains) and SINR-coverage (coverage if the signal-tointerference- and-noise ratio is large). In particular, we prove the existence of the phase transition in the percolation of a spherical Boolean model depending on the grain radius (and, more generally, k-faces in the ČCech simplicial complex, also called clique percolation) on point processes which cluster less than the Poisson process, including determinantal point processes. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation.