Exact solutions for a mean-field Abelian sandpile

We introduce a model for a sandpile, with N sites, critical height N and each site connected to every other site. It is thus a mean-field model in the spin-glass sense. We find an exact solution for the steady state probability distribution of avalanche sizes, and discuss its asymptotics for large N . The publication of a series of articles by P. Bak and collaborators [1, 2] generated a growing interest in the study of certain class of cellular automata models now commonly known as "sandpiles" because of a crude analogy between the dynamical rules and the way sand topples when building a sandpile. The basic motive was not to model the way real sandpiles behave, although it did broaden the audience for granular flow (see e.g. [3, 4, 5]), but to study a phenomenon they coined "self-organized criticality" in which a system reaches a critical state, i.e. a state with no intrinsic length or time scales, without the tuning of an external parameter. This generated a big influx of theoretical,..