Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations

TL;DR

This paper introduces a new continuum line-of-sight percolation model on Poisson-Voronoi tessellations, analyzing phase transitions and critical parameters relevant for telecommunications in random environments.

Contribution

The study develops a novel percolation model on Poisson-Voronoi tessellations, establishing phase laws and providing numerical estimates for critical parameters.

Findings

Existence of 0-1 law, subcritical and supercritical phases.

Development of coarse-graining and stabilization techniques.

Numerical estimates of critical parameters in the planar case.

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 as well as 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.