Critical exponents in percolation via lattice animals

We examine the percolation model by an approach involving lattice animals, divided according to their surface-area-to-volume ratio. Throughout, we work with the bond percolation model in Z. However, the results apply to the site or bond model on any infinite transitive amenable graph with inessential changes. For any given p ∈ (0, 1), two lattice animals with given size are equally likely to arise as the cluster C(0) containing the origin provided that they have the same surface-area-to-volume ratio. For given β ∈ (0,∞), there is an exponential growth rate in the number of edges for the number of lattice animals up to translation that have surface-area-to-volume ratio very close to β. This growth rate f(β) may be studied as a function of β. To illustrate the connection between the percolation model and the combinatorial question of the behaviour of f , note that the probability that the cluster containing the origin contains a large number n of edges is given by