The density conjecture in sandpile models

In sandpile models, various critical densities have been defined, in terms of either the stationary distribution of the driven model in finite volume, or the phase transition in the closed, infinite-volume version of the sandpile. The literature is indecisive and confused about the relations between the various densities or, worse, even fails to distinguish between them. The relations between densities discussed in this proposal play an important role in the practical and philosophical explanation of self-organised criticality. For instance, it is suggested that the critical behaviour of finite-volume driven sandpiles is explained by the fact that the sandpile `mimics the behaviour of infinite sandpiles at the critical point. This idea turns out to be too naive. We suggest that the initial configuration and the so-called local toppling-invariants, play an important role, and that one should be able to formulate and prove the correct relationship, thereby refining the original idea, and settling the dispute in the physics literature. We also propose that the difference between deterministic and stochastic toppling rules will be visible through the relations between the critical densities, and that stochastic and deterministic sandpiles will be shown to have qualitatively different behaviour. In addition, we suggest to initiate the study of slope models, rather than the mathematically more convenient height models.