A factor matching of optimal tail between Poisson processes

TL;DR

This paper constructs a factor perfect matching between two independent Poisson processes in high-dimensional space, achieving an optimal tail decay for the matching distance, advancing understanding of spatial stochastic processes.

Contribution

It introduces a novel method to create a factor perfect matching with optimal tail decay in high-dimensional Poisson processes, combining previous allocation and fractional matching results.

Findings

Matching distance tail decays as $b ext{exp}(-cr^d)$

Constructs a factor perfect matching in high dimensions

Utilizes new combination of allocation and fractional matching theorems

Abstract

Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension $d$ at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., an equivariant measurable function of the point configurations), and with the property that the distance between a configuration point and its pair has a tail distribution that decays as fast as possible, namely, as $b\exp (-cr^d)$ with suitable constants $b,c>0$. Our proof relies on two earlier results: an allocation rule of similar tail for a Poisson point process, and a recent theorem that enables one to obtain perfect matchings from fractional perfect matchings in our setup.