A PROBABILISTIC APPROACH TO THE SITE-PERCOLATION PROBLEM .1. BETHE LATTICES

A simple method, based on the use of conditional probabilities, for deriving the particle cluster distribution is presented for the site-percolation problem. The method is applied to a Bethe lattice (or Cayley tree). Two approximations to the behaviour of the obtained probability distribution in the thermodynamic limit are considered. The first one corresponds to the case of finite clusters, and leads to a binomial-like distribution. The second one allows us to treat the case of the existence of clusters spanning the lattice, and thus to investigate the onset of the percolating cluster. The extrema (maxima) of the corresponding distributions are analytically and numerically analyzed. Well-known results for the Bethe lattice, such as the critical probability, p(c), and the percolation probability, P(proportional to)(p), are obtained.