Approach to criticality in sandpiles

Abstract

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 Z 2 , 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 prove͒ a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the drivendissipative sandpile begins to lose a macroscopic amount of sand to the sink.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

“…Simulations are equally straightforward in the supercritical case, where the above procedure soon leads to an infinite avalanche. As in the BTW case [29], an avalanche will not stop after each site has toppled once, and this will happen in general after O(L) time steps. On the other hand, following avalanches on large lattices until they die is not a viable option at the critical point, because avalanches may not die even after very many time steps.…”

Threshold State and a Conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

“…where the last term removes the extra chips at the sink in order to restore the conditionâ i ρ(z) = deg(z) (recall that ρ andâ i ρ belong to Rec(z), Definition 1). Substituting equation (10) into (11) and then using (12), we find…”

“…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]).…”

Learning about critical phenomena from scribbles and sandpiles

“…It has been argued heuristically that the critical density for the fixed energy model should coincide withσ, 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).…”