Finite clusters in high-density continuous percolation: Compression and sphericality

Abstract

A percolation process inRd is considered in which the sites are a Poisson process with intensity ρ and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other. The distribution of the number of sites in the clusterC of the origin is examined, and related to the geometry ofC. It is shown that when ρ andk are large, there is a characteristic radius λ such that conditionally on |C|=k, the convex hull ofC closely approximates a ball of radius λ, with high probability. When the normal volumek/ρ thatk points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this λ-ball is much greater than the ambient density ρ. For larger normal volumes there is less compression. This can be compared to Bernoulli bond percolation on the square lattice in two dimensions, where an analog of this compression is known not to occur.