Scaling and Crossovers in Diffusion Limited Aggregation

Abstract

We discuss the scaling of characteristic lengths in diffusion limited aggregation clusters in light of recent developments using conformal maps. We are led to the conjecture that the apparently anomalous scaling of lengths is due to one slow crossover. This is supported by an analytical argument for the scaling of the penetration depth of newly arrived random walkers, and by numerical evidence on the Laurent coefficients which uniquely determine each cluster. We find common crossover behavior for the squares of the characteristic lengths and the penetration depth of the form ${N}^{2/D}(\ensuremath{\alpha}+\ensuremath{\beta}{N}^{\ensuremath{-}\ensuremath{\varphi}})$ with $\ensuremath{\varphi}$ in the range $\ensuremath{-}0.3\ifmmode\pm\else\textpm\fi{}0.1$ suggesting that there is a single dominant correction to scaling.