Clustering, percolation and directionally convex ordering of point processes

Abstract

Heuristics indicate that point processes exhibiting clustering of points have larger critical radius $r_c$ for the percolation of their continuum percolation models than spatially homogeneous point processes. It has already been shown, and we reaffirm it in this paper, that the $dcx$ ordering of point processes is suitable to compare their clustering tendencies. Hence, it was tempting to conjecture that $r_c$ is increasing in $dcx$ order. Some numerical evidences support this conjecture for a special class of point processes, called perturbed lattices, which are "toy models" for determinantal and permanental point processes. However, the conjecture is not true in full generality, since 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. Nevertheless, we are able to compare some nonstandard critical radii related, respectively, to the finiteness of the expected number of void circuits around the origin and asymptotic of the expected number of long occupied paths from the origin in suitable discrete approximations of the continuum model. These new critical radii sandwich the "true" one. Surprisingly, the inequalities for them go in opposite directions, which gives uniform lower and upper bounds on $r_c$ for all processes $dcx$ smaller than some given process. In fact, the above results hold under weaker assumptions on the ordering of void probabilities or factorial moment measures only. Examples of point processes comparable to Poisson processes in this weaker sense include determinantal and permanental processes. More generally, we show that point processes $dcx$ smaller than homogeneous Poisson processes exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields, as e.g. $k$-percolation and SINR-percolation.