A three-dimensional cluster problem

Abstract

The expected size of clusler in a three-dimensional Poisson ensemble is discussed. This expected size is a function of one parameter t. There exists a critical value t=t0 at which the expected cluster size becomes infinite and for t>t0 infinite clustors occur. A lower bound t0≥0.0434 and a Monte Carlo estimate t0=0.0889 are obtained. An application of this problem to a percolation process is discussed.