A Factor Matching of Optimal Tail Between Poisson Processes

Timár, Ádám (2023) A Factor Matching of Optimal Tail Between Poisson Processes. COMBINATORICA, 43 (2). pp. 421-427. ISSN 0209-9683

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Abstract

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 b exp(−cr d ) 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.

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