Diffusion-limited aggregation (DLA)

While many 2D cellular automata produce intricate nested shapes, the aggregation models shown here seem to tend to simple limiting shapes. Most likely there are some generalized aggregation models for which this is not the case. And indeed this phenomenon has been seen in other systems with randomness in their underlying rules. An example studied extensively in the 1980s is diffusion-limited aggregation (DLA). The idea of this model is to add cells to a cluster one at a time, and to determine where a cell will be added by seeing where a random walk that starts far from the cluster first lands on a square adjacent to the cluster. An example of the behavior obtained in this model is shown below:

The lack of smooth overall behavior in this case can perhaps be attributed to the global probing of the cluster that is effectively done by each incoming random walk. (See also page 994.)