Number of branches in diffusion-limited aggregates: The skeleton

Abstract

We develop the skeleton algorithm to define the number of main branches ${\mathit{N}}_{\mathit{b}}$ of diffusion-limited aggregation (DLA) clusters. The skeleton algorithm provides a systematic way to remove dangling side branches of the DLA cluster and has successfully been applied to study the ramification properties of percolation. We study the skeleton of comparatively large (\ensuremath{\approxeq}${10}^{6}$ sites) off-lattice DLA clusters in two, three, and four spatial dimensions. We find that initially with increasing distance from the cluster seed the number of branches increases in all dimensions. In two dimensions, the increase in the number of branches levels off at larger distances, indicating a fixed number of ${\mathit{N}}_{\mathit{b}}$=7.5\ifmmode\pm\else\textpm\fi{}1.5 main branches of DLA. In contrast, in three and four dimensions, we find indications that the skeleton continues to ramify as one proceeds from the cluster center outward, and there may not exist a constant number of main branches. Likewise, we find no indication for a fixed ${\mathit{N}}_{\mathit{b}}$ in a study of DLA on the Cayley tree, the limit of ``infinite dimensions.'' In two dimensions, we encounter strong corrections to scaling of logarithmic character, which can help to explain recently reported deviations from self-similar behavior of DLA. \textcopyright{} 1996 The American Physical Society.