Zero dissipation limit in the abelian avalanche model

Details Zero dissipation limit in the abelian avalanche model Zero dissipation limit in the abelian avalanche model

2009-06-17

arxiv.org/abs/0906.3128v3

Mathematics

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33 pages, revised 3rd version, title has been changed

Scientific paper

The discrete height abelian sandpile model was introduced by Bak, Tang & Wiesenfeld as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called dissipative. We give a detailed overview of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show exponential decay of spatial covariances of local observables in the non-zero dissipation regime. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics.

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