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 \v{C}ech 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.Comment: 25 pages, 1 figure. This paper complements arXiv:1111.6017. arXiv admin note: substantial text overlap with arXiv:1105.429

Signal-to-interference ratio percolation for Cox point processes

“…However, [BY10,BY13] also studied the case Φ ⊆ Φ I . These papers investigated certain notions of sub-Poisson point processes.…”

“…[MR96,FM07,BB09] for overviews. After 2010, it has also been extended to various other kinds of point processes, for example sub-Poisson [BY10,BY13], Ginibre and Gaussian zero [GKP16], and Gibbsian [J16, S13]. The case of Gibbsian point processes was also studied earlier, see the references in [J16].…”

“…We will write f (x) = Θ(g(x)) if f (x) = O(g(x)) and g(x) = O(f (x)). In [BY13], a more general notion of SINR graphs was considered, and the results of [DFMMT06] were extended to the case of sub-Poisson point processes in this context. 1.2.…”

On Comparison of Clustering Properties of Point Processes

“…Indeed, Figure 1 hints at ordering of the critical radii of DCX ordered PPs in d = 2. However, as shown in [5], this conjecture is not true in general: there exists a super-Poisson PP with the critical radius equal n)) as replication kernels, having fixed spherical grains of radius r. The replication kernels converge in n, from below and from above in DCX, respectively, to Poisson PPs whose critical radius is depicted by the dashed line.…”

“…Remark 6.1. Even if the finiteness of r c is not clear for Poisson PPs and hence Corollary 6.1 cannot be directly used to prove the finiteness of the critical radii of ν-weakly sub-Poisson PPs, the approach based on void probabilities can be refined, as shown in [5], to conclude the aforementioned property.…”