Geometry of Clusters

We have seen how we can characterize clusters by their mass, s. As p approaches \(p:c\) , the typical cluster size s increases as well as the characteristic cluster diameter. In this chapter we will discuss the geometry of clusters, and by geometry we will mean how the number of sites in a cluster is related to the linear size of the cluster. We will introduce several measures to characterize the spatial extent, the characteristic radius \(R:s\) , of clusters of size s. We will measure \(R:s\) to motivate that it is proportional to \(s^1/D\) , where D is a new exponent characterizing the dimension of clusters. We will demonstrate that the percolation system is characterized by two lengths, the system size L and a characteristic cluster size \(\xi \) , and that the system shows fractal, self-similar scaling when the characteristic length diverges. We develop scaling theories for \(P(s,L)\) for \(p>p_c\) and lay the foundations for a geometrical understanding and description of the spanning cluster.