Scaling properties of diffusion-limited aggregation, the percolation hull, and invasion percolation

We study various properties of the surface of diffusion-limited aggregation (DLA) and invasion percolation clusters using a ''glove algorithm.'' Specifically, we define the l-perimeter to be the set of nonfractal sites with a chemical distance l from a fractal with M sites. We argue that P(M,l), the number of sites of the l-perimeter, should obey a scaling law of the form P(M,l)/l∼f(l/Mf1/d), where f(u)∼uf-d for u→0 and f(u)→const for u→. Simulations of two-dimensional off-lattice DLA clusters, invasion percolation clusters, and percolation hulls-as well as an exact treatment of the Sierpiński gasket-support this scaling form. We find that an analogous scaling form holds for G(M,l), the number of sites in the ''l-glove,'' which is composed of the sites of the l-perimeter accessible to particles of radius l from the exterior. Moreover, we define a hierarchy of ''lagoons'' for the case of loopless fractals as regions that are inaccessible to particles of different sizes. We apply this definition to DLA and find that the lagoon-size distribution in DLA is consistent with a self-similar structure of the aggregate. However, we find even for large lagoons a surprisingly small most probable width of the necks that separate the lagoons from the exterior of the cluster. Small neck widths of large lagoons are consistent with a recently proposed void-neck model for the geometric structure of DLA.