Invariant bipartite random graphs on $\mathbb{R}^d$

Abstract

Suppose that red and blue points occur in $\mathbb{R}^d$ according to two simple point process with finite intensities $λ_{\mathcal{R}}$ and $λ_{\mathcal{B}}$, respectively. Furthermore, let $ν$ and $μ$ be two probability distributions on the strictly positive integers. Assign independently a random number of stubs (half-edges) to each red and blue point with laws $ν$ and $μ$, respectively. We are interested in translation-invariant schemes to match stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law $ν$ or $μ$ depending on its color. Let $X$ and $Y$ be random variables with law $ν$ and $μ$, respectively. For a large class of point processes we show that we can obtain such translation-invariant schemes matching a.s. all stubs if and only if \[ λ_{\mathcal{R}} \mathbb{E}(X)= λ_{\mathcal{B}} \mathbb{E}(Y), \] allowing $\infty$ in both sides, when both laws have infinite mean. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage. For this scheme we give sufficient conditions on $X$ and $Y$ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Häggström and Holroyd.