Studying self-organized criticality with exactly solved models

Abstract

These lecture-notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models. The abelian group structure of the algebra of particle addition operators, the burning test for recurrent states, equivalence to the spanning trees problem are described. The exact solution of the directed version of the model in any dimension, and determination of the exponents for avalanche distribution are explained. The model's equivalence to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model is discussed. For the undirected case, the exact solution in 1-dimension and on the Bethe lattice is briefly described. Known results about the two dimensional case are summarized. Generalization to the abelian distributed processors model is discussed, with the Eulerian walkers model and Manna's stochastic sandpile model as examples. I conclude by listing some still-unsolved problems.