  Consider a Boolean model $\Sigma$ in $\R^d$. The centers are given by a
homogeneous Poisson point process with intensity $\lambda$ and the radii of
distinct balls are i.i.d.\ with common distribution $\nu$. The critical covered
volume is the proportion of space covered by $\Sigma$ when the intensity
$\lambda$ is critical for percolation. Previous numerical simulations and
heuristic arguments suggest that the critical covered volume may be minimal
when $\nu$ is a Dirac measure. In this paper, we prove that it is not the case
at least in high dimension. To establish this result we study the asymptotic
behaviour, as $d$ tends to infinity, of the critical covered volume. It appears
that, in contrast to what happens in the constant radii case studied by
Penrose, geometrical dependencies do not always vanish in high dimension.
