Indistinguishability of cells for the ideal Poisson Voronoi tessellation

TL;DR

This paper proves that in the ideal Poisson Voronoi tessellation, all cells are statistically indistinguishable, using advanced theorems from ergodic theory and point process theory.

Contribution

It demonstrates the indistinguishability of cells in the ideal Poisson Voronoi tessellation and provides an alternative proof of Meyerovitch's theorem.

Findings

Cells in the ideal Poisson Voronoi tessellation are indistinguishable.

Application of Howe-Moore theorem to spatial tessellations.

New proof of Meyerovitch's theorem on Poisson point processes.

Abstract

In this note, we resolve a question of D'Achille, Curien, Enriquez, Lyons, and \"Unel by showing that the cells of the ideal Poisson Voronoi tessellation are indistinguishable. This follows from an application of the Howe-Moore theorem and a theorem of Meyerovitch about the nonexistence of thinnings of the Poisson point process. We also give an alternative proof of Meyerovitch's theorem.