Figure 27.

A large percolation cluster at the critical site concentration $p\simeq 0.3116$ in 3D on a simple cubic lattice. The cluster has been grown using standard techniques [315,316] (see also [310,317] for a generalization of the growth rules). The fractal dimension of percolation clusters is $d_{\mathrm{f}}\simeq 2.53\pm 0.02$ (see, e.g., [318]). The different colors represent the length of the shortest (linear) path connecting a site to the seed of the cluster, the latter is located within the red sites and the blue sites have the longest paths (or chemical distances [310]) from the seed. The cluster contains over a million sites, and the longest chemical distance is 2000 in lattice units (courtesy of Francesco Marini). The fractal dimension of the shortest path is $d_{\min }=1.13077\left(2\right)$ in 2D and $d_{\min }=1.3756\left(6\right)$ in 3D [319]. The shortest paths are essential when dealing with the transport of matter along the connected sites of the cluster.