Approaching Criticality via the Zero Dissipation Limit in the Abelian Avalanche Model
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The discrete height abelian sandpile model was introduced by Bak, Tang, Wiesenfeld and Dhar 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 bulk-dissipative. We provide a detailed study 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 that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. 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. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for \(d > 4\).
KeywordsAbelian sandpile model Abelian avalanche model Toppling probability exponent Burning algorithm Weighted spanning trees Wilson’s algorithm Zero dissipation limit Self-organized criticality
We thank Hermann Thorisson for indicating us reference . E.S. was supported by grants ANR-07-BLAN-0230, ANR-2010-BLAN-0108. For financial support and hospitality, we thank Carleton University, Leiden University, MAP5 lab at Université Paris Descartes, Nijmegen University, Delft University and Centre Emile Borel of Institut Henri Poincaré (part of this work was done during the semester “Interacting Particle Systems, Statistical Mechanics and Probability Theory”).
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