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

TL;DR

This study investigates the possibility of generating isotropic on-lattice DLA clusters using a probabilistic algorithm, revealing inherent anisotropies and the critical conditions under which isotropy can be achieved.

Contribution

The paper introduces a generalized aggregation probability model with a tunable exponent, identifying critical values for pattern isotropy and anisotropy directions in on-lattice DLA.

Findings

In the noiseless limit, clusters show rotated anisotropy patterns.

A critical exponent  separates axial and diagonal anisotropy regimes.

Noise reduction influences the anisotropy direction in DLA patterns.

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 $k^\nu$, was proposed. The exponent $\nu$ has a nonuniversal critical value $\nu_c$, for which the patterns generated in the noiseless limit exhibit the original (axial) anisotropy for $\nu<\nu_c$ and the rotated one (diagonal) for $\nu>\nu_c$. The values $\nu_c = 1.395\pm0.005$ and $\nu_c = 0.82\pm 0.01$ 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 $\nu=2$ (\bogo's rule) is an example where the patterns exhibit the axial anisotropy for small and the diagonal one for large noise reduction.