Is it really possible to grow isotropic on-lattice diffusion-limited aggregates?

Abstract

In a recent paper (Bogoyavlenskiy V A 2002 \JPA \textbf{35} 2533), an algorithm aiming to generate isotropic clusters of the on-lattice diffusion-limited aggregation (DLA) model was proposed. The procedure consists of aggregation probabilities proportional to the squared number of occupied sites ($k^2$). In the present work, we analyzed this algorithm using the noise reduced version of the DLA model and large scale simulations. In the noiseless limit, instead of isotropic patterns, a $45^\circ$ ($30^\circ$) rotation in the anisotropy directions of the clusters grown on square (triangular) lattices was observed. A generalized algorithm, in which the aggregation probability is proportional to $k^ν$, was proposed. The exponent $ν$ has a nonuniversal critical value $ν_c$, for which the patterns generated in the noiseless limit exhibit the original (axial) anisotropy for $ν<ν_c$ and the rotated one (diagonal) for $ν>ν_c$. The values $ν_c = 1.395\pm0.005$ and $ν_c = 0.82\pm 0.01$ were found for square and triangular lattices, respectively. Moreover, large scale simulations show that there are a nontrivial relation between noise reduction and anisotropy direction. The case $ν=2$ (\bogo's rule) is an example where the patterns exhibit the axial anisotropy for small and the diagonal one for large noise reduction.