Percolation in the Boolean model with convex grains in high dimension

Abstract

We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set $K\subset {\mathbb{R}}^d$ with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or $\mathit{\rho }K$ for a further parameter $\mathit{\rho }\in (1,2)$. We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.