Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the absence of percolation in a line-segment based lilypond model

Christian Hirsch

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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.

Résumé

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’.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 127-145.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089263

Digital Object Identifier
doi:10.1214/14-AIHP638

Mathematical Reviews number (MathSciNet)
MR3449297

Zentralblatt MATH identifier
1335.60182

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B4

Keywords
Lilypond model Mass-transport principle Percolation Random geometric graph Sprinkling

Citation

Hirsch, Christian. On the absence of percolation in a line-segment based lilypond model. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 127--145. doi:10.1214/14-AIHP638. https://projecteuclid.org/euclid.aihp/1452089263


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