Abstract
We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are $k$-coverage in the Boolean model (coverage by at least $k$ grains) and SINR-coverage (coverage if the signal-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the \v{C}ech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.
Highlights
Starting with the work of [19], percolation problems on geometric models defined over the Poisson point process have garnered interest among both stochastic geometers and network theorists
In this paper we show that point processes having voids probabilities and moment measures smaller than a Poisson point process, exhibit a non-trivial phase transition in the percolation of their level-set coverage models
Though we focus on the percolation of Boolean models, but as is the wont in the subject we shall extensively use discrete percolation models as approximations
Summary
Starting with the work of [19], percolation problems on geometric models defined over the Poisson point process have garnered interest among both stochastic geometers and network theorists. We relate percolation properites of Boolean models and, more generally, level-sets of additive functionals of point processes, to the more intrinsic properties of the underlying point processes, such as moment measures and void probabilities. It has already been observed in [8] that smaller values of these characteristics indicate less clustering. We observe that the Poisson assumption on point processes is not always preferable in many models of spatial networks In such scenarios, the question of whether clustering or repulsion increases various performance measures (related to some property of the underlying geometric graph) or not arises naturally. We shall use clustering properties of Φ to get bounds on rck(Φ)
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