Fractal properties of the frontier in Poissonian coloring

TL;DR

This paper investigates the fractal nature of the boundary in a Poissonian coloring model, showing that the frontier's Hausdorff dimension lies strictly between $d-1$ and $d$, confirming a conjecture by Aldous.

Contribution

It proves the Hausdorff dimension bounds of the frontier in a Poissonian coloring model, advancing understanding of its fractal properties and confirming a conjecture.

Findings

Frontier has Hausdorff dimension between d-1 and d

Colored regions converge to random closed sets

Answers a conjecture by Aldous about the frontier's dimension

Abstract

We study a model of random partitioning by nearest-neighbor coloring from Poisson rain, introduced independently by Aldous and Preater. Given two initial points in $[0,1]^d$ respectively colored in red and blue, we let independent uniformly random points fall in $[0,1]^d$, and upon arrival, each point takes the color of the nearest point fallen so far. We prove that the colored regions converge in the Hausdorff sense towards two random closed subsets whose intersection, the frontier, has Hausdorff dimension strictly between $d-1$ and $d$, thus answering a conjecture raised by Aldous. However, several topological properties of the frontier remain elusive.