Percolation in invariant Poisson graphs with i.i.d. degrees. (English) Zbl 1254.05181

Summary: Let each point of a homogeneous Poisson process in \(\mathbb{R}^d\) independently be equipped with a random number of stubs (half-edges) according to a given probability distribution \(\mu\) on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution \(\mu\). Leaving aside degenerate cases, we prove that for any \(\mu\) 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 \(\mu\) for the absence and presence of infinite components.


05C80 Random graphs (graph-theoretic aspects)
05C07 Vertex degrees
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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