RANDOM PARTITIONS OF THE PLANE VIA POISSONIAN COLORING AND A SELF-SIMILAR PROCESS OF COALESCING PLANAR PARTITIONS

• 成果类型: Article
• 署名作者: Aldous, David
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1218
• 发表日期: 2018
• 页码: 2000-2037
• 关键词: random tessellations

摘要

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') of the nearest other point z'.