The Poisson Rain Tessellation Under Spatial Expansion and Temporal Transformation

TL;DR

This paper investigates the effects of spatial expansion and temporal transformation on Poisson Rain tessellations, revealing conditions for temporal stationarity and the relationship between temporal transformations and Poisson Rain intensity.

Contribution

It introduces a generalized approach using Poisson Rain to analyze how spatial expansion influences temporal stationarity in cell-division models.

Findings

Exponential spatial expansion is necessary for temporal stationarity.

Temporal transformation is strongly related to Poisson Rain intensity.

The approach generalizes previous models to broader conditions.

Abstract

For cell-division processes in a window, Cowan introduced four selection rules and two division rules each of which stands for one cell-division model. One of these is the area-weighted in-cell model. In this model, each cell is selected for division with a probability that corresponds to the ratio between its area and the area of the whole window. This selected cell is then divided by throwing a (uniformly distributed) point into the cell and drawing a line segment through the point under a random angle with the segment ending at the cell's boundary. For the STIT model which uses both a different selection and a different division rule, Mart\'inez and Nagel showed that for a STIT process {Y(t, W): t > 0} the process {exp(t) Y(exp(t), W): t real} is not only spatially stationary but also temporally. For a continuous-time area-weighted in-cell model it is shown by using the different and generalizing approach of a Poisson Rain that in order to get temporal stationarity it is necessary to have an exponential spatial expansion. For the temporal transformation it is found that there is a strong relation between that transformation and the intensity of the Poisson Rain in time.