Percolation for the stable marriage of Poisson and Lebesgue

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