{"id":341696,"date":"2016-12-27T09:54:54","date_gmt":"2016-12-27T17:54:54","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=341696"},"modified":"2018-10-16T21:25:15","modified_gmt":"2018-10-17T04:25:15","slug":"poisson-matching-2","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/poisson-matching-2\/","title":{"rendered":"Poisson Matching"},"content":{"rendered":"<p>Suppose that red and blue points occur as independent homogeneous Poisson processes in R<em><sup>d<\/sup><\/em>. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions <em>d<\/em> = 1, 2, the matching distance <em>X<\/em> from a typical point to its partner must have infinite <em>d<\/em>\/2-th moment, while in dimensions <em>d<\/em> \u2265 3 there exist schemes where <em>X<\/em> has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance <em>X<\/em> for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in <em>d<\/em> = 1, but far from optimal in <em>d<\/em> \u2265 3. For the problem of matching Poisson points of a single color to each other, in <em>d<\/em> = 1 there exist schemes where <em>X<\/em> has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then <em>X<\/em> must have infinite mean.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Suppose that red and blue points occur as independent homogeneous Poisson processes in Rd. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d = 1, 2, the matching distance X from a typical point to its partner must have infinite d\/2-th moment, while [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Institut Henri Poincar\u00c3\u00a9","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Annales de l'Institut Henri Poincar\u00c3\u00a9, Probabilit\u00c3\u00a9s et 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