  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.
