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 $D=\frac{5}{3}$, whereas $D=2$ for the Eden model, and $D=\frac{4}{3}$ for the $\ensuremath{\eta}=2$ dielectric breakdown model. $D$ is not universal: $D=\frac{7}{4}$ for the two-dimensional triangular lattice. The square and triangular lattices bracket (\ifmmode\pm\else\textpm\fi{}2.5%) Meakin's large off-lattice simulations ($D=1.71$).