Structure of large two-dimensional square-lattice diffusion-limited aggregates: Approach to asymptotic behavior

Efficient algorithms have been used to grow large (4\ifmmode\times\else\texttimes\fi{}${10}^{6}$ site) diffusion-limited aggregation (DLA) clusters on two-dimensional (2D) square lattices. As the clusters grow larger, their envelope grows, from a more or less round shape characteristic of small clusters, through a diamond shape characteristic of clusters containing about ${10}^{5}$ sites, into a cross shape. Results from about 25 clusters indicate that the exponents describing the length l and width w of the four major arms vary continuously with M (the cluster mass) over the range ${10}^{3}$4\ifmmode\times\else\texttimes\fi{}${10}^{6}$. We find that the effective exponent ${\ensuremath{\nu}}_{?}$=dln(l)/dln(M) increases systematically from 0.585 to 0.61 at the highest mass. This may be consistent with a limiting value of (2/3) (as found for uniaxially biased DLA in two dimensions) but only with large corrections to scaling in our range of M. The exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}$=dln(w)/dln(M) decreases systematically, to about 0.48 at M=4\ifmmode\times\else\texttimes\fi{}${10}^{6}$. Our results are consistent with an asymptotic (scaling) fractal geometry for square-lattice DLA but suggest that these fractals are neither self-similar nor homogeneous.