Statistics Seminars: Percolation for the stable marriage of Poisson and Lebesgue
4 October 2006 11:15 in CM107Let~$\Xi$ 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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