Connectivity in Sub-Poisson Networks
Résumé
We consider a class of point processes, which we call sub-Poisson; these are point processes that can be directionally convexly ($dcx$) dominated by some Poisson point process. The $dcx$ order has already been shown useful in comparing various point process characteristics, including Ripley's and correlation functions as well as shot-noise fields generated by point processes, indicating in particular that smaller in the $dcx$ order processes exhibit more regularity (less clustering, less voids) in the repartition of their points. Using these results, in this paper we study the impact of the $dcx$ ordering of point processes on the properties of two continuum percolation models, which have been proposed in the literature to address macroscopic connectivity properties of large wireless networks. As the first main result of this paper, we extend the classical result on the existence of phase transition in the percolation of the Gilbert's graph (called also the Boolean model), generated by a homogeneous Poisson point process, to the class of homogeneous sub-Poisson processes. We also extend a recent result of the same nature for the SINR graph, to sub-Poisson point processes. Finally, we show examples the so-called perturbed lattices, which are sub-Poisson. More generally, perturbed lattices provide some spectrum of models that ranges from periodic grids, usually considered in cellular network context, to Poisson ad-hoc networks, and to various more clustered point processes including some doubly stochastic Poisson ones.
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