Percolation for the stable marriage of Poisson and Lebesgue

Let $\mathrm{\Xi }$ be the set of points (we call the elements of $\mathrm{\Xi }$ centers) of Poisson process in ${\mathbb{R}}^d$, $d\ge 2$, with unit intensity. Consider the allocation of ${\mathbb{R}}^d$ to $\mathrm{\Xi }$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $\alpha\leq 1$\mathrm{.}Weprovethatthereisnopercolationinthesetofclaimedsitesif$\alpha$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $\alpha <1$ is large enough.