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

TL;DR

This paper investigates stable matchings between Poisson point processes in high-dimensional Euclidean space, revealing that unmatched points persist as dimension grows, using the Poisson-weighted infinite tree for analysis.

Contribution

It introduces a novel application of the PWIT to high-dimensional Euclidean stable matching, providing asymptotic results on unmatched points and extending to multi-color and hierarchical metrics.

Findings

Unmatched blue points exist in high dimensions for fixed intensities.

The intensity of unmatched blue points converges to e^{- ho}/(1+ ho) as dimension increases.

Unmatched points persist across all intensities in hierarchical metric settings.

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.