## The Annals of Probability

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

David Aldous

#### 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 $z^{\prime}$.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2000-2037.

Dates
Received: January 2017
Revised: July 2017
First available in Project Euclid: 13 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1528876820

Digital Object Identifier
doi:10.1214/17-AOP1218

Mathematical Reviews number (MathSciNet)
MR3813984

Zentralblatt MATH identifier
06919017

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G57: Random measures

#### Citation

Aldous, David. Random partitions of the plane via Poissonian coloring and a self-similar process of coalescing planar partitions. Ann. Probab. 46 (2018), no. 4, 2000--2037. doi:10.1214/17-AOP1218. https://projecteuclid.org/euclid.aop/1528876820

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