On percolation in one-dimensional stable Poisson graphs

Abstract

Equip each point $x$ of a homogeneous Poisson point 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, Häggström and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph $G$ on the vertex set $\mathcal{P}$. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Häggström and Holroyd [1] 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 $\varepsilon \gt 0$, there exists a distribution $\mu$ such that $\mu(\{1\})\gt 1 -\varepsilon$, but percolation still occurs a.s.