Scaling structure of the growth-probability distribution in diffusion-limited aggregation processes

In nonequilibrium growth such as diffusion-limited aggregation (DLA), the growth-site probability distribution characterizes these growth processes. By solving the Laplace equation numerically, we calculate the growth probability $P_g\mathrm{}\left(x\right)\mathrm{}$ at the perimeter site $x$ of clusters for the DLA and its generalized version called the $\eta$ model, and obtain the generalized dimension $D\left(q\right)\mathrm{}$ and the $f-\alpha$ spectrum proposed by Halsey et al. [Phys. Rev. A 33, 1141 (1986)]. It is found that $D\left(q\right)\mathrm{}$ depends strongly on $q$ and that the $f-\alpha$ spectrum is continuous. Our results suggest that these growth processes cannot be described by a simple scaling theory with a few scaling exponents. This is in clear contrast to the Botet-Jullien model [Phys. Rev. Lett. 55, 1943 (1985)] which yields equilibrium patterns whose $D\left(q\right)\mathrm{}$ is constant. It is also found that the information dimension $D\left(1\right)\mathrm{}$ which represents the properties of the unscreened surface is in good agreement with our recent theory.