FINITE CLUSTERS IN HIGH-DENSITY CONTINUOUS PERCOLATION - COMPRESSION AND SPHERICALITY

A percolation process in R(d) is considered in which the sites are a Poisson process with intensity rho and the bond between each pair of sites is open if and only if the sites are within a fixed distance r of each other. The distribution of the number of sites in the cluster C of the origin is examined, and related to the geometry of C. It is shown that when rho and k are large, there is a characteristic radius lambda such that conditionally on Absolute value of C = k, the convex hull of C closely approximates a bali of radius lambda, with high probability. When the normal volume k/rho that k points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this lambda-ball is much greater than the ambient density rho. 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.