Self-orgainzed criticality and singular diffusion

We suggest that certain open driven systems self-organize to a critical point because their continuum diffusion limits have singularities in the diffusion constants at the critical point. We rigorously establish a continuum limit for a one-dimensional automaton that has this property, and show that certain exponents and the distribution of events are simply related to the order of the diffusion singularity. Numerically, we show that some of these results can be generalized to include a class of sandpile models that are described by a similar, but higher-order, singularity.