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February 2016 On the absence of percolation in a line-segment based lilypond model
Christian Hirsch
Ann. Inst. H. Poincaré Probab. Statist. 52(1): 127-145 (February 2016). DOI: 10.1214/14-AIHP638

Abstract

We prove the absence of percolation in a directed Poisson-based random geometric graph with out-degree $1$. This graph is an anisotropic variant of a line-segment based lilypond model obtained from an asymmetric growth protocol, which has been proposed by Daley and Last. In order to exclude backward percolation, one may proceed as in the lilypond model of growing disks and apply the mass-transport principle. Concerning the proof of the absence of forward percolation, we present a novel argument that is based on the method of sprinkling.

Nous montrons l’absence de percolation dans un graphe géométrique aléatoire orienté de degré sortant 1 construit sur un processus ponctuel de Poisson. Ce graphe est fondé sur une variante anisotropique d’un système de segments qui croissent selon un protocole asymétrique de type ‘lilypond’ proposé par Daley et Last. Pour exclure la percolation en direction des ascendants on peut procéder comme dans le cas d’un système de disques qui croissent selon un protocole de type ‘lilypond’ en utilisant le principe du transport de masse. Concernant la preuve de l’absence de percolation en direction des descendants, nous donnons un nouvel argument à l’aide de la méthode de ‘saupoudrage’.

Citation

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Christian Hirsch. "On the absence of percolation in a line-segment based lilypond model." Ann. Inst. H. Poincaré Probab. Statist. 52 (1) 127 - 145, February 2016. https://doi.org/10.1214/14-AIHP638

Information

Received: 19 November 2013; Revised: 24 July 2014; Accepted: 4 August 2014; Published: February 2016
First available in Project Euclid: 6 January 2016

zbMATH: 1335.60182
MathSciNet: MR3449297
Digital Object Identifier: 10.1214/14-AIHP638

Subjects:
Primary: 60K35
Secondary: 82B4

Rights: Copyright © 2016 Institut Henri Poincaré

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Vol.52 • No. 1 • February 2016
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