Stable Poisson Graphs in One Dimension

TL;DR

This paper studies one-dimensional stable Poisson graphs with degree distributions, proving conditions for the existence of infinite components and analyzing edge lengths, especially when all vertices have degree two.

Contribution

It introduces new results on the existence of infinite components in stable Poisson graphs, including a proof of non-existence for a specific stable matching scheme and simulation-supported conjectures for another.

Findings

No infinite component for the random direction stable matching scheme.

Infinite component existence depends on a finite interval condition.

Simulation evidence supports the conjecture for certain stable matchings.

Abstract

Let each point of a homogeneous Poisson process on $\RR$ independently be equipped with a random number of stubs (half-edges) according to a given probability distribution $\mu$ on the positive integers. We consider schemes based on Gale-Shapley stable marriage for perfectly matching the stubs to obtain a simple graph with degree distribution $\mu$. We prove results on the existence of an infinite component and on the length of the edges, with focus on the case $\mu(\{2\})=1$. In this case, for the random direction stable matching scheme introduced by Deijfen and Meester we prove that there is no infinite component, while for the stable matching of Deijfen, H\"aggstr\"om and Holroyd we prove that existence of an infinite component follows from a certain statement involving a {\em finite} interval, which is overwhelmingly supported by simulation evidence.