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

A percolation process in R(d) is considered in which the sites are a P oisson process with intensity rho and the bond between each pair of si tes 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 o f the origin is examined, and related to the geometry of C. It is show n that when rho and k are large, there is a characteristic radius lamb da such that conditionally on Absolute value of C = k, the convex hull of C closely approximates a bali of radius lambda, with high probabil ity. 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 volu me in this lambda-ball is much greater than the ambient density rho. F or larger normal volumes there is less compression. This can be compar ed to Bernoulli bond percolation on the square lattice in two dimensio ns, where an analog of this compression is known not to occur.