Percolation for the stable marriage of Poisson and Lebesgue

Abstract

Let $\Xi$ be the set of points (we call the elements of $\Xi$ centers) of Poisson process in ${\mathbb{R}}^d$, $d\ge 2$, with unit intensity. Consider the allocation of ${\mathbb{R}}^d$ to $\Xi$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\alpha \le 1$. We prove that there is no percolation in the set of claimed sites if $\alpha$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $\alpha <1$ is large enough.

arXiv commentrevised version (only minor correction since v2), 16 pages, 3 figures

References in corpus (1)

• A stable marriage of Poisson and Lebesgue

Cited by in corpus (2)

• A stable marriage of Poisson and Lebesgue
• Percolation for the stable marriage of Poisson and Lebesgue with random appetites