Approach to criticality in sandpiles

Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands.
Physical Review E (Impact Factor: 2.29). 09/2010; 82(3 Pt 1):031121. DOI: 10.1103/PhysRevE.82.031121
Source: arXiv


A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this "density conjecture" is false when the underlying graph is any of Z2, the complete graph K(n), the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this paper, we substantiate this claim by rigorous proof and extensive simulations. We show that driven-dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. Nevertheless, we do find (and prove) a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the driven-dissipative sandpile begins to lose a macroscopic amount of sand to the sink.

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    • "It has been argued heuristically that the critical density for the fixed energy model should coincide with ¯ σ, and on the square lattice this is indeed numerically the case up to a 0.01% error. However, Fey, Levine, and Wilson[19]numerically disproved this conjecture and showed that the two values are in fact distinct (see[28]however). "

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    • "This is the question lurking beneath the prediction [24] that ζ s = ζ τ : the stationary density of the abelian sandpile should equal the threshold density of the fixed energy sandpile (the density of sand at which it becomes permanently unstable; precise definitions are given below in §1.2–1.8). In [10] [11] the above prediction was refuted on a few simple graphs where ζ τ can be computed exactly, and simulations on the two-dimensional torus Z N × Z N show that ζ τ ≈ 2.125288 differs slightly from ζ s = 2.125000 (the exact evaluation ζ s = 17/8 was recently proved in [23] [16]). Why are these values so close if they are not equal? "
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    ABSTRACT: We prove a precise relationship between the threshold state of the fixed-energy sandpile and the stationary state of Dhar’s abelian sandpile: in the limit as the initial condition s 0 tends to \({-\infty}\) , the former is obtained by size-biasing the latter according to burst size, an avalanche statistic. The question of whether and how these two states are related has been a subject of some controversy since 2000. The size-biasing in our result arises as an instance of a Markov renewal theorem, and implies that the threshold and stationary distributions are not equal even in the \({s_0 \to -\infty}\) limit. We prove that, nevertheless, in this limit the total amount of sand in the threshold state converges in distribution to the total amount of sand in the stationary state, confirming a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle.
    Preview · Article · Feb 2014 · Communications in Mathematical Physics
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    • "One can prove this for a number of specific graphs that have an infinite volume counterpart, for instance C n (where both densities are trivially equal to 1), or, less obvious, for the bracelet graph [18]. The flower graph or the complete graph however do not have an infinite volume counterpart. "
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    ABSTRACT: We discuss various critical densities in sandpile models. The stationary density is the average expected height in the stationary state of a finite-volume model; the transition density is the critical point in the infinite-volume counterpart. These two critical densities were generally assumed to be equal, but this has turned out to be wrong for deterministic sandpile models. We show they are not equal in a quenched version of the Manna sandpile model either. In the literature, when the transition density is simulated, it is implicitly or explicitly assumed to be equal to either the so-called threshold density or the so-called critical activity density. We properly define these auxiliary densities, and prove that in certain cases, the threshold density is equal to the transition density. We extend the definition of the critical activity density to infinite volume, and prove that in the standard infinite volume sandpile, it is equal to 1. Our results should bring some order in the precise relations between the various densities.
    Preview · Article · Nov 2012 · Markov Processes and Related Fields
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