NEAREST-NEIGHBOR AND HARD-SPHERE MODELS IN CONTINUUM PERCOLATION

Consider a Poisson process X in R(d) With density 1. We connect each p oint of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolati on occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolatio n model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ba ll. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general propertie s of percolation models where balls placed at Poisson points are not a llowed to overlap (but are allowed to be tangent). (C) 1996 John Wiley & Sons, Inc.