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

Part of: Special processes Computer system organization Operations research and management science Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2021

Quentin Le Gall Open the ORCID record for Quentin Le Gall [Opens in a new window] ,
Bartłomiej Błaszczyszyn Open the ORCID record for Bartłomiej Błaszczyszyn [Opens in a new window] ,
Élie Cali  and
Taoufik En-Najjary
Quentin Le Gall*
Affiliation:
Orange Labs Networks and Inria—École Normale Supérieure
Bartłomiej Błaszczyszyn*
Affiliation:
Inria—École Normale Supérieure
Élie Cali*
Affiliation:
Orange Labs Networks
Taoufik En-Najjary*
Affiliation:
Orange Labs Networks
*
*Postal address: Orange Labs Networks, Modelling and Statistical Analysis, 44 avenue de la République, CS 50010, 92326 Châtillon Cedex, France.
**Postal address: Centre de recherches Inria de Paris, DYOGENE, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France.
*Postal address: Orange Labs Networks, Modelling and Statistical Analysis, 44 avenue de la République, CS 50010, 92326 Châtillon Cedex, France.
*Postal address: Orange Labs Networks, Modelling and Statistical Analysis, 44 avenue de la République, CS 50010, 92326 Châtillon Cedex, France.
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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.

Keywords

PercolationCox processPoisson–Voronoi tessellationcoarse-graining argumentssimulation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G55: Point processes 60D05: Geometric probability and stochastic geometry
Secondary: 68M10: Network design and communication 90B15: Network models, stochastic
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 510 - 536
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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