Continuum percolation in high dimensions

Consider a Boolean model Sigma in R-d. The centers are given by a homogeneous Poisson point process with intensity. 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. 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.