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2015 On percolation in one-dimensional stable Poisson graphs
Johan Björklund, Victor Falgas-Ravry, Cecilia Holmgren
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Electron. Commun. Probab. 20: 1-6 (2015). DOI: 10.1214/ECP.v20-3958


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.


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Johan Björklund. Victor Falgas-Ravry. Cecilia Holmgren. "On percolation in one-dimensional stable Poisson graphs." Electron. Commun. Probab. 20 1 - 6, 2015.


Accepted: 1 July 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1358.60009
MathSciNet: MR3367900
Digital Object Identifier: 10.1214/ECP.v20-3958

Primary: 60C05
Secondary: 05C70 , 05C80 , 60D05

Keywords: Matching , percolation , Poisson process , random graph

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