The ACO Seminar (2012-2013)

The Abelian sandpile is a chip-firing game on the lattice which can be viewed as a simple deterministic analog to stochastic diffusion processes based on random walks. In contrast to its stochastic counterparts, the sandpile produces striking fractal scaling limits which have long resisted explanation, or even precise description. In this talk, we will discuss a new approach to understanding the fractal behavior of the sandpile, which begins by identifying sandpile limits as solutions of a certain PDE. The heart of our results is a surprising connection between the integer superharmonic functions on the lattice which govern this PDE and Apollonian circle packings of the plane, which allows a characterization of certain fractal solutions produced by the sandpile.