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Poisson matching

Alexander E. Holroyd; Robin Pemantle; Yuval Peres; Oded Schramm

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 266-287
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Holroyd, Alexander E., et al. "Poisson matching." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 266-287. <http://eudml.org/doc/78020>.

@article{Holroyd2009,
abstract = {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.},
author = {Holroyd, Alexander E., Pemantle, Robin, Peres, Yuval, Schramm, Oded},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Poisson process; point process; matching; stable marriage; Gale-Shapley stable marriage; matching distance},
language = {eng},
number = {1},
pages = {266-287},
publisher = {Gauthier-Villars},
title = {Poisson matching},
url = {http://eudml.org/doc/78020},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Holroyd, Alexander E.
AU - Pemantle, Robin
AU - Peres, Yuval
AU - Schramm, Oded
TI - Poisson matching
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 266
EP - 287
AB - 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.
LA - eng
KW - Poisson process; point process; matching; stable marriage; Gale-Shapley stable marriage; matching distance
UR - http://eudml.org/doc/78020
ER -

References

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