The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice , in which sites with at least 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 chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as . 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 . We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.
"Convergence of the Abelian sandpile." Duke Math. J. 162 (4) 627 - 642, 15 March 2013. https://doi.org/10.1215/00127094-2079677