Distribution of growth probabilities for off-lattice diffusion-limited aggregation

We study the distribution n(α,M) of growth probabilities {${\mathit{p}}_{\mathit{i}}$} for off-lattice diffusion-limited aggregation (DLA) for cluster sizes up to mass M=20 000, where ${\mathrm{\alpha }}_{\mathit{i}}$==-${\mathit{p}}_{\mathit{i}}$/logM. We find that for large α, log n(α,M)∝-${\mathrm{\alpha }}^{\gamma }$/${\log }^{\mathrm{\delta }}$M, with γ=2±0.3 and δ=1.3±0.3. One consequence of this form is that the minimum growth probability ${\mathit{p}}_{\min }$(M) obeys the asymptotic relation log${\mathit{p}}_{\min }$(M)∼-(logM${)}^{(\gamma +1+\mathrm{\delta })/\gamma }$. We find evidence for the existence of a well-defined crossover value ${\mathrm{\alpha }}^{\mathrm{*}}$ such that only the rare configurations of DLA contribute to n(α,M) for α>${\mathrm{\alpha }}^{\mathrm{*}}$, while both rare and typical DLA configurations contribute for α${\mathrm{\alpha }}^{\mathrm{*}}$.