Geometric properties of Poisson matchings

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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.

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Holroyd, A. E. (2011). Geometric properties of Poisson matchings. Probability Theory and Related Fields, 150(3), 511–527.

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