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 $l_{\perp }$, 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 ${\nu }_{?}$∼(2/3) and ${\nu }_{\perp }$∼(1/2). From the finite-size scaling analysis we derive the expression D=1+(1-${\nu }_{?}$)/${\nu }_{\perp }$ 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.