On percolation in Poisson graphs

TL;DR

This paper investigates percolation in Poisson graphs formed by pairing stubs on a Poisson process, demonstrating that infinite connectivity can occur even with certain restrictive degree distributions.

Contribution

It proves that percolation can happen even with degree distributions supported on odd integers and highly concentrated on degree one.

Findings

Percolation occurs with distributions supported on odd integers.

Percolation can occur even when most vertices have degree one.

Infinite components can form under highly restrictive degree distributions.

Abstract

Equip each point $x$ of a homogeneous Poisson process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable multi-matching scheme introduced by Deijfen, H\"aggstrom and Holroyd (2012), we pair off edge stubs in a series of rounds to form the edge set of an infinite component $G$ on the vertex set $\mathcal{P}$. In this note, we answer questions of Deijfen, Holroyd and Peres (2011) and Deijfen, H\"aggstr\"om and Holroyd (2012) on percolation (the existence of an infinite connected component) in $G$. We prove that percolation may occur a.s. even if $\mu$ has support over odd integers. Furthermore, we show that for any $\varepsilon>0$ there exists a distribution $\mu$ such that $\mu(\{1\})>1-\varepsilon$ such that percolation still occurs a.s..