Diffusion limited aggregation as a Markovian process. Part I: bond-sticking conditions

Abstract

Cylindrical lattice Diffusion Limited Aggregation (DLA), with a narrow width N, is solved using a Markovian matrix method. This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The matrix is then used to find the weights of the steady state growing configurations and the rate of approaching this steady state stage. The former are then used to find the average upward growth probability, the average steady-state density and the fractal dimensionality of the aggregate, which is extrapolated to a value near 1.64.