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

In a recent paper (Bogoyavlenskiy V A 2002 J. PhYs. A: Math. Gen. 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 analysed 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 degrees (30 degrees) 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(v), was proposed. The exponent v has a nonuniversal critical value v, for which the patterns generated in the noiseless limit exhibit the original (axial) anisotropy for v < v(c) and the rotated one (diagonal) for v > v(c). The values v(c) = 1.395 +/- 0.005 and v(c) = 0.82 +/- 0.01 were found for square and triangular lattices, respectively. Moreover, large-scale simulations show that there is a nontrivial relation between the noise reduction and anisotropy direction. The case v = 2 (Bogoyavlenskiy's rule) is an example where the patterns exhibit the axial anisotropy for small and the diagonal one for large noise reduction.