How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

We study the fractal structure of diffusion-limited aggregation (DLA) clusters on a square lattice by extensive numerical simulations (with clusters having up to 108 particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has a nontrivial box-counting fractal dimension which corresponds to the overall growth rate, which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around the one-dimensional backbone of the cluster.