Description
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 a configuration point and its pair has a tail distribution that decays as fast as possible in magnitude, namely, as b exp(−crd) with suitable constants b, c > 0.
This settles the most difficult case of unicolored/bicolored
deterministic/randomized such matching problems. Our proof relies on two earlier results: an allocation ("land-division") rule of similar tail for a Poisson point process by Markó and T., based on the Ajtai-Komlós-Tusnády algorithm, and a recent breakthrough result of Bowen, Kun and Sabok that enables one to obtain perfect matchings from fractional perfect matchings under suitable conditions.
For Zoom access please contact Miklos Rasonyi (rasonyi.miklos[a]renyi.hu).