A probabilistic approach to the site-percolation problem

A simple method, based on the use of conditional probabilites, to derive the particle cluster distribution is presented for the site-percolation problem on standard lattices. 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, while the second one allows us to deal with 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 distribution are analytically and numerically analyzed.