Random partitions of the plane via Poissonian coloring, and a self-similar process of coalescing planar partitions

TL;DR

This paper studies a random partitioning process of the plane generated by Poissonian coloring, proving convergence of empirical measures to Lebesgue measures and introducing a novel self-similar coalescing process of planar partitions.

Contribution

It provides a rigorous proof of measure convergence in a Poissonian coloring model and introduces a new self-similar coalescing process of planar partitions.

Findings

Normalized empirical measures converge to Lebesgue measures

Introduction of a self-similar coalescing process

Partial topological properties remain open

Abstract

Plant differently colored points in the plane, then let random points ("Poisson rain") fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar process of coalescing planar partitions. In this process, sets $A(z)$ in the partition are associated with Poisson random points $z$, and the dynamics are as follows. Points are deleted randomly at rate $1$, when $z$ is deleted, its set $A(z)$ is adjoined to the set $A(z^\prime)$ of the nearest other point $z^\prime$.