Absence of percolation for Poisson outdegree-one graphs

TL;DR

This paper proves that certain Poisson outdegree-one graphs do not percolate under specific assumptions, confirming a conjecture for the line-segment model and introducing new models satisfying these conditions.

Contribution

It establishes general conditions (Shield and Loop assumptions) that guarantee absence of percolation in Poisson outdegree-one graphs, including solving a conjecture for the line-segment model.

Findings

No percolation occurs under the Shield and Loop assumptions.

The line-segment model does not percolate, confirming a previous conjecture.

New geometric navigation models also satisfy these assumptions.

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 a conjecture by Daley et al. on the absence of percolation for the line-segment model. 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.