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 approximately f(l/M1/df), where f(u) approximately u(-df) for u --> 0 and f(u) --> const for u --> infinity. Simulations of two-dimensional off-lattice DLA clusters, invasion percolation clusters, and percolation hulls-as well as an exact treatment of the Sierpinski 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.