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
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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.@en