Continuum percolation in high dimensions

Jean-Baptiste Gouéré, Regine Marchand

Published 2011-08-31, updated 2013-03-20Version 2

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.