Electronic Journal of Probability

Stable Poisson Graphs in One Dimension

Maria Deijfen, Alexander Holroyd, and Yuval Peres

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Let each point of a homogeneous Poisson process on R 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, Häggström and Holroyd we prove that existence of an infinite component follows from a certain statement involving a <em>finite</em> interval, which is overwhelmingly supported by simulation evidence

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 44, 1238-1253.

Accepted: 6 July 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Poisson process random graph degree distribution matching percolation

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Deijfen, Maria; Holroyd, Alexander; Peres, Yuval. Stable Poisson Graphs in One Dimension. Electron. J. Probab. 16 (2011), paper no. 44, 1238--1253. doi:10.1214/EJP.v16-897. https://projecteuclid.org/euclid.ejp/1464820213

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