Using stochastic conformal mappings, we study the effects of anisotropic perturbations on diffusion-limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map $m$ preferred directions for growth to a distribution on the unit circle, which is a periodic function with $m$ peaks in $[-\pi ,\pi )$ such that the angular width $\sigma$ of the peak defines the “strength” of anisotropy $\varkappa ={\sigma }^{-1}$ along any of the $m$ chosen directions. The two parameters $(m,\varkappa )$ map out a parameter space of perturbations that allows a continuous transition from DLA (for small enough $\varkappa$) to $m$ needlelike fingers as $\varkappa \to \infty$. We show that at fixed $m$ the effective fractal dimension of the clusters $D(m,\varkappa )$ obtained from mass-radius scaling decreases with increasing $\varkappa$ from $D_{\text{DLA}}\simeq 1.71$ to a value bounded from below by $D_{\min }=\frac{3}{2}$. Scaling arguments suggest a specific form for the dependence of the fractal dimension $D(m,\varkappa )$ on $\varkappa$ for large $\varkappa$ which compares favorably with numerical results.

• Received 20 July 2003

DOI:https://doi.org/10.1103/PhysRevE.69.061403