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.
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) 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conclude that the term self-organised criticality is somewhat confusing, since the tuning of parameters in a model has been replaced by the careful choice of a suitable model.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 show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1 f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects. We study a general Bak-Tang-Wiesenfeld-type automaton model of self-organized criticality in which the toppling conditions depend on local height, but not on its gradient. We characterize the critical state, and determine its entropy for an arbitrary finite lattice in any dimension. The two-point correlation function is shown to satisfy a linear equation. The spectrum of relaxation times describing the approach to the critical state is also determined exactly.