On a Continuum Percolation Model

By Mathew D. Penrose.

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 cluster density, or free energy. Also, we derive a formula for the probability that an arbitraty 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 Poissoon particle is isolated, given that it lies in a finite cluster, approaches unity.

Advances in Applied Probability 23, 536-556 (1991).