Percolation for the stable marriage of Poisson and Lebesgue

Details Percolation for the stable marriage of Poisson and Lebesgue Percolation for the stable marriage of Poisson and Lebesgue

2005-11-07

arxiv.org/abs/math/0511186v3

Stochastic Processes and their Applications, v. 117, p. 514-525, 2007

Mathematics

Probability

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

Scientific paper

10.1016/j.spa.2006.09.002

Let $\Xi$ be the set of points (we call the elements of $\Xi$ centers) of Poisson process in $\R^d$, $d\geq 2$, with unit intensity. Consider the allocation of $\R^d$ to $\Xi$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\alpha\leq 1$. 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.

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