Percolation for the stable marriage of Poisson and Lebesgue

Let ΞΞ be the set of points (we call the elements of ΞΞ centers) of a Poisson process in RdRd, d≥2d≥2, with unit intensity. Consider the allocation of RdRd to ΞΞ which is stable in the sense of the Gale–Shapley marriage problem and in which each center claims a region of volume α≤1α≤1. We prove that there is no percolation in the set of claimed sites if αα is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if α<1α<1 is large enough.