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

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