%0 Journal Article
%T Percolation for the stable marriage of Poisson and Lebesgue
%A Marcelo Ventura Freire
%A Serguei Popov
%A Marina Vachkovskaia
%J Mathematics
%D 2005
%I arXiv
%R 10.1016/j.spa.2006.09.002
%X 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.
%U http://arxiv.org/abs/math/0511186v3