Bipartite stable Poisson graphs on R

TL;DR

This paper studies bipartite graphs formed from Poisson point processes on the real line, showing that when all vertices have degree 2, the resulting graph almost surely lacks an infinite component, contrasting with one-color models.

Contribution

It introduces a bipartite stable matching model on the real line and proves the absence of infinite components for degree-2 vertices, highlighting a key difference from one-color models.

Findings

Graphs with degree-2 vertices have no infinite component.

Bipartite and one-color models exhibit different percolation behaviors.

Simulation results explore other degree distributions.

Abstract

Let red and blue points be distributed on $\mathbb{R}$ according to two independent Poisson processes $\mathcal{R}$ and $\mathcal{B}$ and let each red (blue) point independently be equipped with a random number of half-edges according to a probability distribution $\nu$ ($\mu$). We consider translation-invariant bipartite random graphs with vertex classes defined by the point sets of $\mathcal{R}$ and $\mathcal{B}$, respectively, generated by a scheme based on the Gale-Shapley 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 two-color 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.