February 2024 A shape theorem for exploding sandpiles
Ahmed Bou-Rabee
Author Affiliations +
Ann. Appl. Probab. 34(1A): 714-742 (February 2024). DOI: 10.1214/23-AAP1976

Abstract

We study scaling limits of exploding Abelian sandpiles using ideas from percolation and front propagation in random media. We establish sufficient conditions under which a limit shape exists and show via a family of counterexamples that convergence may not occur in general. A corollary of our proof is a simple criterion for determining if a sandpile is explosive; this strengthens a result of Fey, Levine and Peres (J. Stat. Phys. (2010) 138 143–159).

Acknowledgments

Thank you to Charles K. Smart for motivating, helpful discussions during this project. Thank you to Lionel Levine for several inspiring conversations and for suggesting this question. Thank you to Dylan Airey for asking if exploding sandpiles have a limit shape.

Citation

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Ahmed Bou-Rabee. "A shape theorem for exploding sandpiles." Ann. Appl. Probab. 34 (1A) 714 - 742, February 2024. https://doi.org/10.1214/23-AAP1976

Information

Received: 1 February 2021; Revised: 1 July 2022; Published: February 2024
First available in Project Euclid: 28 January 2024

MathSciNet: MR4696289
zbMATH: 07829154
Digital Object Identifier: 10.1214/23-AAP1976

Subjects:
Primary: 60K35
Secondary: 60K37

Keywords: Abelian sandpile , Bootstrap percolation , discrete Laplacian , discrete reaction-diffusion , Growth model

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.34 • No. 1A • February 2024
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