Poisson allocations with bounded connected cells

TL;DR

This paper presents methods to partition the plane into equal-volume, bounded, connected cells based on a Poisson point process, with optimal tail bounds on cell diameter and properties ensuring either positive separation or finite intersection with bounded regions.

Contribution

It introduces two novel translation-invariant constructions for Poisson allocations with bounded connected cells, achieving optimal tail bounds and distinct spatial interaction properties.

Findings

Cells have diameter tail bound P(D>r)<c/r

Two variants: one with positive distance between cells

One with finite intersection with any bounded region

Abstract

Given a homogenous Poisson point process in the plane, we prove that it is possible to partition the plane into bounded connected cells of equal volume, in a translation-invariant way, with each point of the process contained in exactly one cell. Moreover, the diameter $D$ of the cell containing the origin satisfies the essentially optimal tail bound $P(D>r)<c/r$. We give two variants of the construction. The first has the curious property that any two cells are at positive distance from each other. In the second, any bounded region of the plane intersects only finitely many cells almost surely.