ABSTRACT
Percolation is a random process. Therefore, different percolation lattices will contain clusters of different sizes and shapes. In order to discuss their average properties, one must study the statistics of these clusters. This is done by studying the number of clusters with s sites per lattice site, n
cluster numbers for the simple cases of one dimension, lattice animals and on Bethe lattices. These exact solutions first suggest a simple functional form, and this is then generalized to the scaling form of Eq. (33). The moments of the cluster numbers yield the strength P of the infinite cluster and the mean cluster size S, and scaling implies relations among the corresponding critical exponents and amplitudes. Series expansions and Monte Carlo simulations are used to check these theoretical predictions.
