NONOPTIMALITY OF CONSTANT RADII IN HIGH DIMENSIONAL CONTINUUM PERCOLATION

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 in sufficiently high dimension.