Ballistic aggregation displays self-organized criticality

Abstract

Consider the convex hull of a collection of disjoint open discs with radii $1/2$. The boundary of the convex hull consists of a finite number of line segments and arcs. Randomly choose a point in one of the arcs in the boundary so that the density of its distribution is proportional to the arc measure. 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 small side branches and occasional macroscopic side branches. 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). Our results indicate that the size of a branch obeys a power law, as expected of avalanches in sandpile models and similar ``catastrophies'' in other SOC models.