Arkiv för Matematik

Percolation in invariant Poisson graphs with i.i.d. degrees

Maria Deijfen, Olle Häggström, and Alexander E. Holroyd

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Abstract

Let each point of a homogeneous Poisson process in ℝd independently be equipped with a random number of stubs (half-edges) according to a given probability distribution μ on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution μ. Leaving aside degenerate cases, we prove that for any μ there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on μ for the absence and presence of infinite components.

Article information

Source
Ark. Mat., Volume 50, Number 1 (2012), 41-58.

Dates
Received: 6 February 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907158

Digital Object Identifier
doi:10.1007/s11512-010-0139-8

Mathematical Reviews number (MathSciNet)
MR2890343

Zentralblatt MATH identifier
1254.05181

Rights
2011 © Institut Mittag-Leffler

Citation

Deijfen, Maria; Häggström, Olle; Holroyd, Alexander E. Percolation in invariant Poisson graphs with i.i.d. degrees. Ark. Mat. 50 (2012), no. 1, 41--58. doi:10.1007/s11512-010-0139-8. https://projecteuclid.org/euclid.afm/1485907158


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