Convergence of the Abelian sandpile

Understand

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible.

• From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$.
• However, little has been proved about the appearance of the stable configurations.
• We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n \to \infty$.