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, 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.
The abelian sandpile has a much longer history: it arose in statistical physics as a model of self-organized criticality' (SOC) @cite_5 @cite_10 . The dichotomy between stabilizing and exploding configurations arose in the course of a debate about whether SOC does or does not involve tuning a parameter to a critical value @cite_7 @cite_9 . Without reopening that particular debate, we view the stabilizing exploding dichotomy as a topic with its own intrinsic mathematical interest. An example of its importance can be seen in the partial differential equation for the scaling limit of the abelian sandpile on @math , which relies on a classification of certain quadratic' sandpiles according to whether they are stabilizing or exploding @cite_0 .
@cite_7 @cite_9 @cite_0 @cite_5 @cite_10
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We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable. Key-words: Self-organized criticality, abelian sandpile model, activated random walkers, stabilizability. AMS classification: 60K35 (primary), 60G60 (secondary) A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the Abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the Abelian sandpile model at stationarity. We investigate the nature of the self-organised critical behaviour in the Abelian sandpile model and in the Bak Sneppen evolution model. We claim that in either case, the self-organised critical behaviour can be explained by the careful choice of the details of the model: they are designed in such a way that the models are necessarily attracted to the critical point of a conven- tional parametrised equilibrium system. In the case of the Abelian sandpile we prove this connection to conventional criticality rigorously in one dimension, and provide evidence for a similar result in higher dimensions. In the case of the Bak Sneppen evolution model, we give an overview of the current results, and explain why these results support our claim. We conclude that the term self-organised criticality is somewhat confusing, since the tuning of parameters in a model has been replaced by the careful choice of a suitable model. Viewed as such, we can hardly call this critical behaviour spontaneous. We consider the Abelian sandpile model (ASM) on the square lattice with a single dissipative site (sink). Particles are added one by one per unit time at random sites and the resulting density of particles is calculated as a function of time. We observe different scenarios of evolution depending on the value of initial uniform density (height) h0. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density ρc(h0), the system changes its behavior and relaxes exponentially to the stationary state of the ASM with density ρs. Considering initial heights −1≤h0≤4, we observe a dramatic decrease of the difference ρc(h0)−ρs when h0 is zero or negative. In parallel with the ASM, we consider the conservative fixed energy sandpile (FES). The extensive Monte Carlo simulations show that the threshold density ρth(h0) of the FES converges rapidly to ρs for h0<1.