Asymptotic shape of small cells

TL;DR

This paper investigates the asymptotic shapes of small cells in a specific Poisson line tessellation, revealing degenerate shapes for small area and non-unique shapes for small perimeter, supported by simulation results.

Contribution

It provides new insights into the asymptotic geometry of small cells in Poisson line tessellations with two directional atoms, contrasting with large cell behavior.

Findings

Small cells with tiny area tend to have degenerate shapes.

Small cells with tiny perimeter do not have a unique asymptotic shape.

Simulation studies support the theoretical findings.

Abstract

A stationary Poisson line tessellation is considered whose directional distribution is concentrated on two different atoms with some positive weights. The shape of the typical cell of such a tessellation is studied when its area or its perimeter tends to zero. In contrast to known results where the area or the perimeter tends to infinity, it is shown that the asymptotic shape of cells having small area is degenerate. Again in contrast to the case of large cells, the asymptotic shape of cells with small perimeter is not uniquely determined. The results are accompanied by a large scale simulation study.