Asymptotic shape of small cells

Mareen Beermann, Claudia Redenbach, Christoph Thaele

Published 2012-11-23, updated 2013-09-13Version 2

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.