A Factor Matching of Optimal Tail Between Poisson Processes

Ádám Timár*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays as fast as possible in magnitude, namely, as bexp (- crd) with suitable constants b, c> 0 . This settles the most difficult version of such matching problems: bicolored (versus unicolored) and deterministic (versus randomized). Our proof relies on two earlier results: an allocation (“land-division”) rule of similar tail for a Poisson point process by Markó and the author, and a recent breakthrough result of Bowen, Kun and Sabok that enables one to obtain perfect matchings from fractional perfect matchings under suitable conditions.

Original languageEnglish
Pages (from-to)421-427
Number of pages7
Issue number2
Publication statusPublished - 1 Apr 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s).

Other keywords

  • Allocation rule
  • Factor matching
  • Land division
  • Poisson point process


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