Electronic Communications in Probability

On percolation in one-dimensional stable Poisson graphs

Johan Björklund, Victor Falgas-Ravry, and Cecilia Holmgren

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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.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 50, 6 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C70: Factorization, matching, partitioning, covering and packing 05C80: Random graphs [See also 60B20]

Poisson process Random graph Matching Percolation

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Björklund, Johan; Falgas-Ravry, Victor; Holmgren, Cecilia. On percolation in one-dimensional stable Poisson graphs. Electron. Commun. Probab. 20 (2015), paper no. 50, 6 pp. doi:10.1214/ECP.v20-3958. https://projecteuclid.org/euclid.ecp/1465320977

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