Convergence of the Abelian sandpile

Abstract

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 topple, distributing one 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 partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n\to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.