The divisible sandpile at critical density

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 at most 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 biLaplacian Gaussian field.

arXiv comment34 pages, to appear in Annales Henri Poincare

References in corpus (6)

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Cited by in corpus (6)

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• Exact sampling and fast mixing of Activated Random Walk
• Convergence of the random Abelian sandpile
• Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous
• Odometers of Divisible Sandpile Models: Scaling Limits, iDLA and Obstacle Problems. A Survey