A simple model of kinetic growth process is diffusion-limited aggregation (DLA), which consists of particles in Brownian motion that "stick" together in a square lattice. In the 1D case, particles are added in random positions with the same value of height, which increases at each step. For the 2D case, other subtleties are involved; a particle is released from a randomly selected initial location within a circle of radius to move randomly. If the particle moves to a location contiguous to an occupied site, it is added to the cluster. The particle continues to move until it is captured or moves a distance away from the original position in the circle. When the distance is varied, the cluster can be develops into a fractal pattern, as described in the paper by Witten and Sander.
T. A. Witten Jr, L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981)
Huang, Yi-Bin; Somasundaran, P., Phys. Rev. A 36, 4518 (1987)