Finite dimension and particle heterogeneous DLAs

We study heterogeneous Diffusion Limited Aggregates (DLAs) i.e. those formed by a mixture, in different proportions, of 4-legged and 2-legged particles. We fixed the total number of particles, let the proportions vary, and computed their finite dimension, a recent addition to the list of "fractal" dimensions. At one extreme, when all particles are 4-legged, the corresponding DLAs are complex, fractal structures whose appearance resembles very much that of the DLAs that occur in Nature. At the other extreme, when almost all particles are 2-legged, the DLAs lose much of their complexity and acquire long rectilinear stretches so that their appearance resembles more and more the structure of the underlying lattice. We expected the complexity in between would decrease monotonically, and this would be reflected in the finite dimension of the corresponding DLAs. However, the finite dimension first increases and then, when the proportion of 4-legged to 2-legged particles is about 30 to 70, starts decreasing. In the paper, we study and explain the mechanisms behind this unexpected, counter-intuitive behaviour.