Variability of the form and of the harmonic measure for small off-off-lattice diffusion-limited aggregates

The harmonic measure on off-lattice diffusion-limited (DL) aggregates is defined in detail. The formulation requires the continuum Laplacian and the regularization of divergences in the density of the measure due to folds between atoms. An ultraviolet cutoff is needed for numerical simulations, but in this off-off-lattice formulation, the cutoff can be varied at will. The cutoff’s effects on the distribution of the Hölders α is studied. We find that the right-hand tail of the distribution of α is very dependent on the lower lattice cutoff. However, the cutoff does not effect the transformations that collapse the distributions corresponding to clusters of different sizes. The shapes of off-lattice DL aggregates giving rise to extremely small minimal harmonic measures are illustrated. We discuss the notion of the smallest harmonic measure in an ensemble of clusters, and also the scaling properties of the distribution of α on small clusters.