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

TL;DR

This paper proves that a specific directed geometric graph derived from a lilypond model does not exhibit percolation, using novel probabilistic methods to analyze anisotropic growth and connectivity.

Contribution

It introduces a new proof technique based on sprinkling to establish the absence of forward percolation in an anisotropic lilypond model.

Findings

No percolation occurs in the directed Poisson-based lilypond model.

Backward percolation is excluded via the mass-transport principle.

A novel argument based on sprinkling is used to prove the absence of forward percolation.

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.