Abelian Sandpiles

Year of origin: 1987

Posted online: 2018-10-23 19:53:11Z by Henrik Shahgholian275

Cite as: G-181023.1

  • Combinatorics
  • Analysis of PDEs
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Problem's Description

Abelian Sandpile Model (ASM) is a lattice growth model for configurations of chips distributed on vertices of $\mathbb Z^d$. A vertex carrying at least $2d$ chips topples giving a single chip to all its $2d$ lattice neighbours, and losing $2d$ chips itself. If there are no sites with more than $2d-1$ chips, the process terminates. For any finite non-negative initial configuration of chips, subsequently toppling all sites with at least $2d$ chips, the process terminates in finite steps. This process is abelian in the sense that the final configuration of chips is independent of the order of toppling. The model originates in the work of Bak-Tang-Wiesenfeld [2].

 

The limiting shape of the two-dimensional ASM on $\mathbb{Z}^2$ with i - SciLag
The limiting shape of the two-dimensional ASM on $\mathbb{Z}^2$ with initial $10\ 000 \ 000$ particles placed at the origin. Sites of $\mathbb{Z}^2$ having $0,1,2,$ or $3$ number of chips are coloured by black, purple, red, and blue respectively.

 

As the number of particles tend to infinity the ASM has a uniques scaling limit which tends to a free boundary problems with facets. See [3].

  1. ArticleIs an originDiscrete Balayage and Boundary Sandpile

    Journal d'Analyse Mathématique, 2017arXiv

  2. ArticleIs an originSelf-organized criticality: An explanation of the 1/f noise

    Physical Review Letters 59, 381, 1987

  3. ArticleIs an originConvergence of the Abelian sandpile

    Duke Mathematical Journal 162, 627-642, 2013

  1. Open Convexity of Abelian Sandpile

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  • Edited: (subproblem(s) edited ) at 2018-10-24 14:02:39Z

  • Edited: (references edited ) at 2018-10-24 08:25:11Z View this version

  • Edited: (references edited subproblem(s) edited ) at 2018-10-24 07:32:31Z View this version

  • Created at: 2018-10-23 19:53:11Z View this version