Abstract
The Abelian Sandpile Model (Dhar 1990) is a discrete diffusion process, defined on graphs, which serves as the standard model of self-organized criticality. One is allowed to add sand particles on the nodes of the graph such that each node can stably hold at most some bounded number of particles. The particles flow through the graph as a consequence of surpassing the node capacities, until they reach a special sink node possessing infinite capacity. These simple dynamics give rise to a very interesting Markovian system. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent. We identify a small set of key graph properties that guarantee polynomial bounds on transience classes of the sandpile families satisfying them. These properties governing the speed of sandpile diffusion process are volume growth parameters, boundary regularity type properties and non-empty interior type constraints.This generalizes a previous result by Babai and Gorodezky (2007), in which they establish polynomial bounds on the $n\times n$ grid. Indeed the properties we show are based on ideas extracted from their proof as well as the continuous analogs in the theory of harmonic functions.
Highlights
The Abelian sandpile model(ASM) is a type of discrete diffusion process defined on graphs
The Abelian Sandpile Model (Dhar 1990) is a discrete diffusion process, defined on graphs, which serves as the standard model of self-organized criticality
We identify a small set of key graph properties that guarantee polynomial bounds on transience classes of the sandpile families satisfying them
Summary
The Abelian sandpile model(ASM) is a type of discrete diffusion process defined on graphs. The case of a random adversary, where particles are added in a uniformly random manner across the graph, yields to a simple coupon collector type argument This results in polynomial bounds on the expected time to recurrence (as previously noted in [4]). The problem acquires a distinctly potential theoretic flavor In this scenario, our goal is to add particles so as to avoid a recurrent state for as long as possible. Our goal is to add particles so as to avoid a recurrent state for as long as possible This problem was highlighted by Babai and Toumpakari [4] where they define this number of particles as the transience class of the sandpile. Our study is aimed at trying to understand the underlying potential theoretic properties which characterize polynomial transience
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