On the Distribution function of area and perimeter for planar poisson line process

TL;DR

This paper derives the full distribution function of areas and perimeters for partitions of the plane created by a Poisson line process, extending previous moment calculations to a comprehensive probabilistic description.

Contribution

It introduces a method to compute the entire distribution function of partition components in a Poisson line process, using Riccati equations and Laplace transforms.

Findings

Derived the distribution function of areas and perimeters for Poisson line partitions

Reduced kinetic equations to Riccati form via Laplace transform

Extended previous moment calculations to full distribution functions

Abstract

The challenges of examining random partitions of space are a significant class of problems in the theory of geometric transformations. Richard Miles calculated moments of areas and perimeters of any order (including expectation) of the random division of space in 1972. In the paper we calculate whole distribution function of random divisions of plane by poisson line process. The idea is to interpret a random polygon as the evolution of a segment along a moving straight line. In the plane example, the issue connected with an infinite number of parameters is overcome by considering a secant line. We shall take into account the following tasks: {\textbf 1.} On the plane, a random set of straight lines is provided, all shifts are equally likely, and the distribution law is of the form $F(\varphi).$ What is the area distribution of the partition's components? {\textbf 2.} On the plane, a random set of points is marked. Each point $A$ has an associated area of attraction, which is the collection of points in the plane to which the point $A$ is the nearest of the designated ones. In the first problem, the density of moved sections adjacent to the line allows for the expression of the balancing ratio in kinetic form. Similarly, you can write the perimeters kinetic equations. We will demonstrate how to reduce these equations to the Riccati equation using the Laplace transformation in this paper.