Self-similarity of harmonic measure on DLA

Abstract

The right-hand side of the f{hook}(α) curve of the harmonic measure on DLA is undefined. This does not necessarily imply that the harmonic measure and the DLA geometry are not self-similar. We show for off-lattice DLA that the right-hand tail satisfies a different rescaling rule. This Cauchy rescaling is compatible with self-similarity. The analysis is done on off-off-lattice DLA in which both the Brownian motion and the Laplace equation are off-lattice. The cluster sizes range between 32 and 50 000 atoms. The square lattice used to numerically estimate the Laplacian potential introduces a lower cutoff on the spatial resolution of this potential. We find a dependence of the right tail of the distribution of Hölders α on this ultraviolet cutoff. Whereas the shape of the tail does depend on this ultraviolet lattice cutoff, the applicability of the collapse rules do not. © 1992.