Continuum line-of-sight percolation on Poisson–Voronoi tessellations

Abstract

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

Chase–escape in dynamic device-to-device networks

“…By a slight abuse of notation, we also denote by X i = (X i,t ) t≥0 the trajectory of device X i in S. For any pair of devices X i and X j , we then consider the set of contact times Z(X i , X j ) = t ≥ 0: |X i,t − X j,t | < r and X i,t , X j,t are on the same street as the times where the devices are on the same street and r-close together, where r > 0 is another parameter in the model, the connectivity threshold. Let us note that the constraint that contact times require the devices to be on the same street can be seen as a very strict shadowing assumption [17].…”