ABSTRACT
So far, we have looked only on the distribution of cluster sizes. We now turn to discuss the geometry of the clusters. We first look at the ‘surface’ of a cluster, i.e. its ‘perimeter’. We then introduce the cluster’s linear size, via its radius. In Section 1.3 we saw that the incipient infinite cluster has an internal fractal geometry, reflected by the dependence of its density on the length scale. We now discuss similar fractal relations between the radii of finite clusters at p
show that these results also hold for p$p c , for length scales small compared with the
correlation length ". For larger length scales one observes a crossover to different behaviours. Similar scaling arguments are then applied to quantify the description of Section 1.3 and to relate the fractal dimension D to other exponents (Eq. (54)). These discussions also introduce hyperscaling.
