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).