On a continuum percolation model

Abstract

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.