A factor matching of optimal tail between Poisson processes

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., an equivariant measurable function of the point configurations), and with the property that the distance between a configuration point and its pair has a tail distribution that decays as fast as possible, namely, as $b\exp (-cr^d)$ with suitable constants $b,c>0$. Our proof relies on two earlier results: an allocation rule of similar tail for a Poisson point process, and a recent theorem that enables one to obtain perfect matchings from fractional perfect matchings in our setup.