Numerical simulations of diffusion - limited aggregation on the torus

The results of numerical simulations of diffusion-limited aggregation on the torus are presented. The usual random walk was generalised by allowing the particle to perform jumps of length equal to s lattice spacings, s 3 1. Patterns with periodic structure were obtained. Recently there has been an increasing interest in the study of irreversible kinetic processes leading to the formation of fractal patterns. A simple stochastic model for the formation of clusters of particles in two-dimensional space was proposed by Witten and Sander (1981, 1983). In their model, called diffusion-limited aggregation (DLA), a single particle walks randomly on the square lattice until it reaches another particle ('seed'), located usually in the centre of the lattice. Next, a new particle initiates its random walk. If the particle contacts the cluster (now built of two particles) it is incorporated into the cluster and the cluster grows. This process is repeated many times and leads to ramified structures possessing remarkable scaling properties (see figure 1, where a cluster of 3000 particles is shown). For example, the number N of Figure 1. Typical aggregate obtained from 3000 particles by means of usual DLA on the torus.