Fractal properties of the frontier in Poissonian coloring

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.