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.
Maria Deijfen. Olle Häggström. Alexander E. Holroyd. "Percolation in invariant Poisson graphs with i.i.d. degrees." Ark. Mat. 50 (1) 41 - 58, April 2012. https://doi.org/10.1007/s11512-010-0139-8