Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Absence of percolation for Poisson outdegree-one graphs

David Coupier, David Dereudre, and Simon Le Stum

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Abstract

A Poisson outdegree-one graph is an oriented graph based on a Poisson point process such that each vertex has only one outgoing edge. The paper focuses on the absence of percolation for such graphs. Our main result is based on two assumptions. The Shield assumption ensures that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch of the graph merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. Several models satisfy these general assumptions and inherit in consequence the absence of percolation. In particular, we solve in Theorem 3.1 a conjecture by Daley et al. on the absence of percolation for the line-segment model (Conjecture 7.1 of (Probab. Math. Statist. 36 (2016) 221–246), discussed in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 127–145) as well). In this planar model, a segment is growing from any point of the Poisson process and stops its growth whenever it hits another segment. The random directions are picked independently and uniformly on the unit sphere. Another model of geometric navigation is presented and also fulfills the Shield and Loop assumptions.

Résumé

Nous considérons des graphes orientés dont l’ensemble des sommets est donné par un processus ponctuel de Poisson et tels que chaque sommet admette une et une seule arête sortante. Le résultat principal de ce papier est l’absence de percolation pour de tels graphes satisfaisant deux hypothèses. L’hypothèse Shield stipule que l’état du graphe localement ne dépend que de son voisinage, tandis que l’hypothèse Loop prétend que toute branche orientée du graphe échoue sur un cycle dès qu’un ensemble (fini) de sommets bien choisis est ajouté au processus de Poisson. Plusieurs modèles intéressants satisfont ces deux hypothèses générales et, par conséquent, ne percolent pas. Nous résolvons ainsi dans Theorem 3.1 une conjecture de Daley et al. sur l’absence de percolation pour le “line-segment model” (Conjecture 7.1 de (Probab. Math. Statist. 36 (2016) 221–246), également discutée dans (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 127–145)). Dans ce modèle bidimensionnel, un segment pousse depuis chaque point du processus de Poisson jusqu’à ce qu’il heurte un autre segment, stoppant ainsi sa croissance. Les directions dans lesquelles poussent les segments sont choisies uniformément sur la sphère et indépendamment les unes des autres. Enfin, un autre modèle dit de navigation est présenté et satisfait aussi les hypothèses Shield et Loop.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1179-1202.

Dates
Received: 20 February 2018
Revised: 12 April 2019
Accepted: 29 April 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345634

Digital Object Identifier
doi:10.1214/19-AIHP998

Mathematical Reviews number (MathSciNet)
MR4076780

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Stochastic geometry Geometric random graphs Percolation Mass transport principle Lilypond model Line-segment model

Citation

Coupier, David; Dereudre, David; Le Stum, Simon. Absence of percolation for Poisson outdegree-one graphs. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1179--1202. doi:10.1214/19-AIHP998. https://projecteuclid.org/euclid.aihp/1584345634


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