On a steep mountain slope, a single falling rock sometimes triggers an avalanche, and sometimes gets stuck. It all depends on the state of the slope. A similar transition from quiescence to a cascade was studied using a cellular automaton — a simplified model where collective behavior is governed by Stochastic modeling. This automaton concept was proposed by John von Neumann back in the mid-20th century.
By varying conditions, you can observe two regimes. Activity either dies out, or expands in waves and fills the entire space. But there is a narrow critical interval where the system neither damps nor runs away into limitless growth. Here, avalanches reign. A single small excitation triggers a chain reaction — a cascade of events. The spatial pattern of activity becomes fractal: equally complex whether viewed from afar or up close. Thus the study helps discern that critical slope. The edge where even a tiny pebble unleashes an avalanche.
This balancing act on the edge of order and chaos is called self-organized criticality. A simple example is a sandpile. By adding grains on top, the slope maintains a critical angle on its own. And each new grain can set off an avalanche.