Stable matchings in high dimensions via the Poisson-weighted infinite tree

Abstract: We consider the stable matching of two independent Poisson processes in $\mathbb{R}^d$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Indeed, if the ratio of red to blue intensities is $\rho$ then the intensity of unmatched blue points converges to $e^{-\rho}/(1+\rho)$ as $d\to\infty$. We also establish analogous results for certain multi-color variants. Our proof uses stable matching on the Poisson-weighted infinite tree (PWIT), which can be analyzed via differential equations. The PWIT has been used in many settings as a scaling limit for models involving complete graphs with independent edge weights, but we believe that this is the first rigorous application to high-dimensional Euclidean space. Finally, we analyze the asymmetric matching problem under a hierarchical metric, and show that there are unmatched points for all intensities.