Abstract
The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field.
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Communicated by Anton Bovier.
L. Levine was supported by NSF DMS-1243606 and a Sloan Fellowship.
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Levine, L., Murugan, M., Peres, Y. et al. The Divisible Sandpile at Critical Density. Ann. Henri Poincaré 17, 1677–1711 (2016). https://doi.org/10.1007/s00023-015-0433-x
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DOI: https://doi.org/10.1007/s00023-015-0433-x