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July 2018 Random partitions of the plane via Poissonian coloring and a self-similar process of coalescing planar partitions
David Aldous
Ann. Probab. 46(4): 2000-2037 (July 2018). DOI: 10.1214/17-AOP1218

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 $z^{\prime}$.

Citation

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David Aldous. "Random partitions of the plane via Poissonian coloring and a self-similar process of coalescing planar partitions." Ann. Probab. 46 (4) 2000 - 2037, July 2018. https://doi.org/10.1214/17-AOP1218

Information

Received: 1 January 2017; Revised: 1 July 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919017
MathSciNet: MR3813984
Digital Object Identifier: 10.1214/17-AOP1218

Subjects:
Primary: 60D05
Secondary: 60G57

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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