Bias-free simulation of diffusion-limited aggregation on a square lattice

We identify sources of systematic error in traditional simulations of the Witten-Sander model of diffusion-limited aggregation (DLA) on a square lattice. Based on semi-analytic solutions of the walk-to-line and walk-to-square first-passage problems, we develop an algorithm that reduces the simulation bias to below $10^{-12}$. We grow clusters of $10^8$ particles on $65536\times 65536$ lattices. We verify that lattice DLA clusters inevitably grow into anisotropic shapes, dictated by the anisotropy of the aggregation process. We verify that the fractal dimension evolves from the continuum DLA value, $D=1.71$, for small disk-shaped clusters, towards Kesten's bound of $D=3/2$ for highly anisotropic clusters with long protruding arms.