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Electronic Journal of Probability 18, 72 (2013) 1-20
Clustering and percolation of point processes
Bartlomiej Blaszczyszyn1, D. Yogeshwaran

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.
1:  INRIA Paris-Rocquencourt - DYOGENE