Clustering and percolation of point processes

Details Clustering and percolation of point processes Clustering and percolation of point processes

2011-12-09

arxiv.org/abs/1112.2227v1

Mathematics

Probability

21 pages, 2 figures. This paper complements arXiv:1111.6017 and is a reduced version of arXiv:1105.4293

Scientific paper

We show that simple, stationary point processes of a given intensity on $\mR^d$, having void probabilities and factorial moment measures smaller than those of a homogeneous Poisson point process of the same intensity, admit uniformly non-degenerate lower and upper bounds on the critical radius $r_c$ for the percolation of their continuum percolation models. Examples are negatively associated point processes and, more specifically, determinantal point processes. More generally, we show that point processes $dcx$ smaller than a homogeneous Poisson point processes (for example perturbed lattices) exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields of these point processes. Examples of such models are $k$-percolation and SINR-percolation models. Our study is motivated by heuristics and numerical evidences obtained for perturbed lattices, indicating that point processes exhibiting stronger clustering of points have larger $r_c$. Since the suitability of the $dcx$ ordering of point processes for comparison of clustering tendencies was known, it was tempting to conjecture that $r_c$ is increasing in the $dcx$ order. However the conjecture is not true in full generality as one can construct a Cox point process with degenerate critical radius $r_c=0$, that is $dcx$ larger than a given homogeneous Poisson point process.

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