Poisson matching
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 266-287

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/2th moment, while in dimensions d≥3 there exist schemes where X 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 X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d≥3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.

Supposons que des points rouges et bleus évoluent suivant des processus de Poisson homogènes indépendants dans ℝd. Nous nous intéressons à des procédés invariants par translation appariant de manière bijective les points rouges et les points bleus. En dimensions d=1, 2, quelque soit le procédé considéré, la distance d'appariement (matching distance) X entre un point typique et son partenaire possède nécessairement un d/2-ème moment infini. En revanche, en dimensions d≥3 il existe des procédés pour lesquels X a des moments exponentiels finis. Le «mariage stable» de Gale-Shapley est un procédé naturel, obtenu en appariant une à une les paires mutuellement les plus proches. L'un des principaux résultats de cet article est que dans le cas de ce procédé, la distance d'appariement X est majorée par une loi de puissance. Une minoration en loi de puissance est également vérifiée. En particulier, le mariage stable est essentiellement optimal (en terme de queue de distribution) en dimension d=1, mais il est loin d'être optimal en dimensions d≥3. Dans le cas du problème qui consiste à apparier des points d'une seule couleur issus d'un processus de Poisson, en dimension d=1 il existe des procédés pour lesquels X a des moments exponentiels finis. Par contre, si l'on demande en plus que l'appariement soit une fonction déterministe du processus ponctuel, alors X a nécessairement une moyenne infinie.

DOI : https://doi.org/10.1214/08-AIHP170
Classification:  60D05,  60G55,  05C70
Keywords: Poisson process, point process, matching, stable marriage
     author = {Holroyd, Alexander E. and Pemantle, Robin and Peres, Yuval and Schramm, Oded},
     title = {Poisson matching},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {266-287},
     doi = {10.1214/08-AIHP170},
     zbl = {1175.60012},
     mrnumber = {2500239},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_1_266_0}
Holroyd, Alexander E.; Pemantle, Robin; Peres, Yuval; Schramm, Oded. Poisson matching. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 266-287. doi : 10.1214/08-AIHP170. http://www.numdam.org/item/AIHPB_2009__45_1_266_0/

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