Here are clusters grown after 512 steps, starting with a single seed in the upper-left corner. The color cycles every 86 steps. Shown are five different rules, depending on how many neighbors you need in the cluster in order for your site to be added. In other words, we are trying to color the cells of the cubic lattice with three different colors, so that cells are colored differently from each of their six neighbors. However, here there are still neighboring pairs of the same color, and these pairs form loops analogous to smoke rings or cosmic strings.
Horizontal dominoes are red or green, depending on parity, and vertical ones are blue or yellow.
Here the frozen region, in which the coloring is forced into a periodic pattern, has been removed. They seem to be roughly circular, although there are theoretical reasons to believe that they might be some sort of superellipse instead. Note also the bands of different colors, which correspond to contours in the height function. Quantum random walks Quantum random walks consist of a diffusive process which is unitary rather than stochastic. Here are some random walks on a two-dimensional lattice.