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.

考虑在维度d至少为 3 的欧几里得空间中单位强度的两个独立的泊松点过程。我们在两个点集之间构造一个完美匹配，这是一个因子（即，与平移交换的点配置的可测量函数），并且具有两个匹配配置点之间的距离具有尾部分布的特性，该尾部分布的大小尽可能快地衰减，即，具有适当常数的\ (b\exp (-cr^d)\) 0\)。这解决了此类匹配问题的最困难版本：双色（与单色）和确定性（与随机）。我们的证明依赖于两个早期结果：Markó 和作者提出的泊松点过程的相似尾部分配（“土地划分”）规则，以及 Bowen、Kun 和 Sabok 最近的突破性结果，使人们能够获得完美匹配来自适当条件下的分数完美匹配。