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 {\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.

12 pages, 2 figures, acroread recommended for figure display

References in corpus (1)

• Growth Rates and Explosions in Sandpiles

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• Convergence of the random Abelian sandpile
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• Sandpiles on the heptagonal tiling
• Scaling limit for a long-range divisible sandpile
• Deterministic Abelian Sandpile and square-triangle tilings
• Dynamic dimensional reduction in the Abelian sandpile
• The spectrum of the abelian sandpile model
• Perturbed divisible sandpiles and quadrature surfaces