We consider a class of point processes (pp), which we call {\em sub-Poisson};
these are pp that can be directionally-convexly (dcx) dominated by some
Poisson pp. 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 pp, 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 pp 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 pp, to the class of
homogeneous sub-Poisson pp. We also extend a recent result of the same nature
for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the
so-called perturbed lattices 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 pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an
extended version see http://hal.inria.fr/inria-00497707 version
