Diffusion-limited aggregation near the percolation threshold

Diffusion-limited aggregation (DLA) and dielectric breakdown (DB) models have been used to simulate growth controlled by a Laplacian field on a square lattice network. A fraction ƒ (near the percolation threshold ƒc) of the bonds had a high conductivity (equal to 1), while the others had a low conductivity, equal to R. We used 10-5 < R < 1 for DLA and R = 10-8 for DB. We find crossover from growth on an incipient percolation cluster, with fractal dimensionality D ≅ 1.3, for small length scales, to that on a uniform substrate (D ≅ 1.7), for a large length scales. The crossover length behaves as LR ≈-a, with the crossover exponent a ≅ 0.25. The results were using a scaling theory.