North-HollandPhysica A 187 (1992)551-574 ~ ~ Growth and form in the zero-noise limit of discrete Laplacian growth processes with inherent surface tension I. The square lattice M.T. Batchelor a and B.I. Henry b a Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra ACT 2601, Australia b Department of Applied Mathematics, University of New South Wales, Kensington NSW 2033, Australia Received 6 April 1992 Laplacian growth models that include surface tension in a lowest approximation are sim- ulated on the square lattice in the deterministic zero-noise limit. The models include the dielectric breakdown model with exponent r/and a generalized diffusion-limited aggregation (DLA) model with local sticking probability s = a3-B, where B is the number of neigh- bouring aggregate sites and c~ 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 tran- sition 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 (c~ = 1 ). We further extend this formalism to describe the collapse of the needle and subsequent tip-splitting in the generalized DLA model. I. Introduction Over the past decade there has been remarkable success in reproducing the ge- ometrical complexity of non-equilibrium growth processes with simple computer models that employ random growth rules (for reviews, see e.g., refs. [ 1-3 ] ). De- spite this success, there is no standard theory to account for the behaviour of the computer models themselves. This is largely because each computer grown clus- ter is just one Monte Carlo sample in a typically enormous space of all possible samples. In a recent paper [4] we introduced and implemented a determinis- tic scheme for realizing the zero-noise limit of discrete non-equilibrium growth processes. A major advantage of this scheme is that important differences be- tween cluster morphologies obtained from different models, or from the same 0378-4371/92/$ 05.00 Q 1992-Elsevier Science Publishers B.V. All rights reserved