Geometric Properties of Poisson Matchings

TL;DR

This paper investigates the geometric and combinatorial properties of translation-invariant matchings between independent Poisson point processes in various dimensions, revealing existence and non-existence results for minimal and crossing-free matchings.

Contribution

It establishes new existence results for minimal and finitely intersecting matchings in different dimensions and explores open problems about crossing-free matchings in the plane.

Findings

Existence of locally minimal length matchings in 1D and 3D+

Existence of matchings with finite edges intersecting bounded sets in 2D+

Open problem on crossing-free matchings in 2D

Abstract

Suppose that red and blue points occur as independent Poisson processes of equal intensity in R^d, and that the red points are matched to the blue points via straight edges in a translation-invariant way. We address several closely related properties of such matchings. We prove that there exist matchings that locally minimize total edge length in d=1 and d>=3, but not in the strip R x [0,1]. We prove that there exist matchings in which every bounded set intersects only finitely many edges in d>=2, but not in d=1 or in the strip. It is unknown whether there exists a matching with no crossings in d=2, but we prove positive answers to various relaxations of this question. Several open problems are presented.