Duke Mathematical Journal

Convergence of the Abelian sandpile

Wesley Pegden and Charles K. Smart

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The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice Zd, 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. 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. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

Article information

Duke Math. J., Volume 162, Number 4 (2013), 627-642.

First available in Project Euclid: 15 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 35R35: Free boundary problems


Pegden, Wesley; Smart, Charles K. Convergence of the Abelian sandpile. Duke Math. J. 162 (2013), no. 4, 627--642. doi:10.1215/00127094-2079677. https://projecteuclid.org/euclid.dmj/1363355689

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