Active Zone of Growing Clusters: Diffusion-Limited Aggregation and the Eden Model

Growth is described in terms of an active zone defined as the region where new particles join the existing cluster. This zone is characterized by the probability $P(r,N)\mathrm{}\mathrm{dr}$ that the $N\mathrm{th}$ particle is deposited a distance $r$ from the center of mass. Our Monte Carlo simulations for two-dimensional diffusion-limited aggregation and the Eden process show that, for large $N$, $P(r,N)={\left[{\left(2\mathrm{}\pi \right)}^{\frac{1}{2}}{\xi }_N\right]}^{-1\mathrm{}}\exp [-\frac{{(r-{\bar{r}}_N)}^2}{2{\xi }_N^2}]$, where ${\bar{r}}_N\sim N^{\nu }$ and ${\xi }_N\sim N^{\bar{\nu }}$. We find $\bar{\nu }<\nu$ indicating the presence of two distinctly diverging lengths in these processes.