Occupancy-Probability Scaling in Diffusion-Limited Aggregation

A continuous-time random-walk theory of diffusion-limited aggregation yields perimeter occupancy probabilities. Scaling relates the fractal dimension D to the cluster-tip occupancy probabilities. These agree with the analytic probabilities near cusps of a lattice-symmetric array of traps. On a two-dimensional square lattice D=5/3, whereas $D=2$ for the Eden model, and D=4/3 for the η=2 dielectric breakdown model. D is not universal: D=7/4 for the two-dimensional triangular lattice. The square and triangular lattices bracket (±{}2.5%) Meakin's large off-lattice simulations ($D=1.71$).