Bipartite Stable Poisson graphs on R

Let red and blue points be distributed on R according to two independent Poisson processes R and B and let each red (blue) point independently be equipped with a random number of half-edges according to a probability distribution ν (μ). We consider translation-invariant bipartite random graphs with vertex classes defined by the point sets of R and B, respectively, generated by a scheme based on the GaleShapley stable marriage for perfectly matching the half-edges. Our main result is that, when all vertices have degree 2 almost surely, then the resulting graph does not contain an infinite component. The twocolor model is hence qualitatively different from the one-color model, where Deijfen, Holroyd and Peres have given strong evidence that there is an infinite component. We also present simulation results for other degree distributions.