Exponentially small growth probabilities in diffusion-limited aggregation

Abstract

We present numerical results that suggest the existence of exponentially small growth probabilities in diffusion-limited aggregation (DLA). Based on this observation, we provide a novel quantitative analysis for the qth moment Z(q,L) of the growth probability distribution for a DLA cluster of linear size L. We demonstrate that Z(q,L)\ensuremath{\propto}${L}^{1\mathrm{\ensuremath{-}}q}$ for 0<q\ensuremath{\le}1, and show that the finite-size correction becomes substantial below ${q}_{\ifmmode\times\else\texttimes\fi{}}$\ensuremath{\propto}${L}^{1\mathrm{\ensuremath{-}}D}$. Furthermore, we find a lower bound for the divergence of Z(q,L) for q<0. Our results consolidate the picture of DLA as a self-organized critical state, and support quantitatively the arguments by Blumenfeld and Aharony concerning the divergence of Z(q,L).