Convergence of the Abelian sandpile

Details Convergence of the Abelian sandpile Convergence of the Abelian sandpile

2011-04-30

arxiv.org/abs/1105.0111v2

Mathematics

Analysis of PDEs

12 pages, 2 figures, acroread recommended for figure display

Scientific paper

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 $n \to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

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