Diverging length scales in diffusion-limited aggregation

Applying finite-size scaling analysis to diffusion-limited aggregation (DLA) clusters grown in finite width strips on a square lattice we find that l? and l_, the cluster lengths along and perpendicular, respectively, to the direction of growth, diverge as, respectively, where N is the number of particles in the cluster. We find numerically that? (2/3) and _ (1/2). From the finite-size scaling analysis we derive the expression D=1+ (1-? ) /_ for the fractal dimension D of DLA clusters on a square lattice. The value D (5/3) predicted from this relation agrees with the expected result.