On the homogeneous planar Poisson point process

Abstract

An n-figure is defined as an ordered subset of n distinct points of the plane. With probability 1, the random aggregate of n-figures generated by a homogeneous planar Poisson point process conforms to various ergodic distributions, explicit forms of which are given. The article concludes with several applications. One relates to certain natural random polygonal tessellations generated by the process, another to the minimal and maximal disk concentrations within a region when equal disks are centered at the points of the process.