A Factor Matching of Optimal Tail Between Poisson Processes

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