Critical Behavior of the Sandpile Model as a Self-Organized Branching Process

Abstract

Kinetic equations, which explicitly take into account the branching nature of sandpile avalanches, are derived. The dynamics of the sandpile model is described by the generating functions of a branching process. The real space renormalization group approach to the critical behavior of this model is formulated in terms of the generating functions describing in detail toppling processes inside the blocks. The obtained height probabilities and critical exponent $\tau =1.248\mathrm{}$ are in excellent agreement with the corresponding exact values.