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 ν?{~}(2/3) and ν_⊥{~}(1/2). From the finite-size scaling analysis we derive the expression D=1+(1-ν?)/ν_⊥ 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.