Growth and form in the zero-noise limit of discrete Laplacian growth processes with inherent surface tension. I. The square lattice

Laplacian growth models that include surface tension in a lowest approximation are simulated on the square lattice in the deterministic zero-noise limit. The models include the dielectric breakdown model with exponent η and a generalized diffusion-limited aggregation (DLA) model with local sticking probability s = α^3-B, where B is the number of neighbouring aggregate sites and α is a parameter. We identify two morphological transitions in the zero-noise limit for these models as the effective surface tension is increased; (i) a transition from a stable needle staircase to tip-splitting and (ii) a transition from axial growth to diagonal growth. We use a stationary contour approximation and conformal mapping methods to obtain theoretical estimates for the step lengths in the stable needle staircase for the DLA model with zero surface tension (α = 1). We further extend this formalism to describe the collapse of the needle and subsequent tip-splitting in the generalized DLA model.