Geometric Properties of Poisson Matchings

Suppose that red and blue points occur as independent Poisson processes of equal intensity in R, 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 × [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.