Self-organized critical state of sandpile automaton models

Avalanches, Sandpiles and Tutte Decomposition

“…This theorem is equivalent to the description of R as the set of those stable configurations which do not contain forbidden subconfigurations (Dhar, 1990). For a subset W of V , a subconfiguration {h i , i ∈ W } is called forbidden if…”

“…Theorem 3 (Dhar, 1990). The set R is a fundamental domain for the lattice L generated by the vectors δ i , i.e.…”

“…The following properties (Dhar, 1990;Bjorner and Lovasz, 1991) play the principal role in the dynamics of our models.…”

“…Avalanche models vs. sandpile models. An Abelian sandpile model (Dhar, 1990) is defined in the same way as the Abelian avalanche model, for an integer matrix ∆, except the values of h i are integer, time is discrete, and the loading rate v is random, with the probability v i to add 1 to the value h i at every time step.…”

“…Statistical properties of an important class of these models, Abelian sandpiles (Dhar, 1990) and Abelian avalanches (Gabrielov, 1992), can be investigated analytically due to an Abelian group acting on the phase space. It is shown that the distribution of avalanches in a discrete, stochastic Abelian sandpile model is identical to the distribution of avalanches in a continuous, deterministic Abelian avalanche model with the same redistribution matrix and loading rate vector.…”