Apollonian structure in the Abelian sandpile

TL;DR

This paper mathematically characterizes the fractal structures in the Abelian sandpile's scaling limit, linking it to quadratic growths of integer-superharmonic functions and providing a new understanding of its complex patterns.

Contribution

It introduces a novel mathematical framework connecting the sandpile's fractal patterns to quadratic growths of integer-superharmonic functions, explaining the structure's origin.

Findings

Proves the sandpile PDE admits fractal solutions.

Establishes a connection between sandpile patterns and quadratic growths.

Provides a rigorous mathematical explanation for the fractal nature.

Abstract

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.