Ballistic aggregation displays self-organized criticality

The following model was proposed by Stefan Steinerberger.
Consider the convex hull of a collection of disjoint open discs with
identical radii. The boundary of the convex hull consists of a finite
number of line segments and arcs. Randomly (uniformly) choose a point
in one of the arcs in the boundary. Attach a new disc at the chosen
point so that it is outside of the convex hull and tangential to its
boundary. Replace the original convex hull with the convex hull of all
preexisting discs and the new disc. Continue in the same manner.

Simulations show that disc clusters form long, straight, or slightly
curved filaments with many small side branches and occasional
macroscopic side branches. For a large number of discs, the shape of
the convex hull is either an equilateral triangle or a quadrangle.
Side branches play the role analogous to avalanches in sandpile
models, one of the best-known examples of self-organized criticality
(SOC).  Simulation and theoretical results indicate that the size of a
branch obeys a power law, as expected of avalanches in sandpile models
and similar ``catastrophes'' in other SOC models.