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

TL;DR

This paper studies how random graphs formed from Poisson point processes with i.i.d. degrees can have finite or infinite components, depending on the matching scheme and degree distribution, with specific results for a Gale-Shapley inspired scheme.

Contribution

It proves the existence of matching schemes that produce only finite or infinite components for any degree distribution, and provides conditions for the Gale-Shapley scheme to have infinite components.

Findings

Existence of schemes with only finite components for any degree distribution.

Existence of schemes with infinite components for any degree distribution.

Conditions under which the Gale-Shapley scheme produces infinite components.

Abstract

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 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 that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.