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-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the Čech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. 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, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.

25 pages, 1 figure. This paper complements arXiv:1111.6017. arXiv admin note: substantial text overlap with arXiv:1105.4293

References in corpus (7)

• Random geometric complexes
• Strict inequalities of critical values in continuum percolation
• Connectivity in Sub-Poisson Networks
• Perturbing the hexagonal circle packing: a percolation perspective
• Strict inequalities of critical probabilities on Gilbert's continuum percolation graph
• Clustering, percolation and directionally convex ordering of point processes
• Continuum Percolation for Gaussian zeroes and Ginibre eigenvalues

Cited by in corpus (7)

• On the topology of random complexes built over stationary point processes
• On comparison of clustering properties of point processes
• Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees
• Continuum Percolation and Stochastic Epidemic Models on Poisson and Ginibre Point Processes
• Gilbert's disc model with geostatistical marking
• Percolation phase transitions for the SIR model with random powers
• Thresholds for vanishing of `Isolated' faces in random Čech and Vietoris-Rips complexes