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

Maria Deijfen Department of Mathematics, Stockholm University Olle Häggström Department of Mathematics, Chalmers University of Technology Alexander E. Holroyd Microsoft Research, 1 Microsoft Way, Redmond, WA, U.S.A.

Optimization and Control Probability mathscidoc:1701.27003

Arkiv for Matematik, 50, (1), 41-58, 2010.2
Let each point of a homogeneous Poisson process in ℝ^{$d$}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.
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  title={Percolation in invariant Poisson graphs with i.i.d. degrees},
  author={Maria Deijfen, Olle Häggström, and Alexander E. Holroyd},
  booktitle={Arkiv for Matematik},
Maria Deijfen, Olle Häggström, and Alexander E. Holroyd. Percolation in invariant Poisson graphs with i.i.d. degrees. 2010. Vol. 50. In Arkiv for Matematik. pp.41-58.
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