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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-organizedcriticality. When the model is modified to allow grains to disappear on eachtoppling, it is called bulk-dissipative. We provide a detailed study of acontinuous height version of the abelian sandpile model, called the abelianavalanche model, which allows an arbitrarily small amount of dissipation totake place on every toppling. We prove that for non-zero dissipation, theinfinite volume limit of the stationary measure of the abelian avalanche modelexists and can be obtained via a weighted spanning tree measure. We show thatin the whole non-zero dissipation regime, the model is not critical, i.e.,spatial covariances of local observables decay exponentially. We then study thezero dissipation limit and prove that the self-organized critical model isrecovered, both for the stationary measure and for the dynamics. We obtainrigorous bounds on toppling probabilities and introduce an exponent describingtheir scaling at criticality. We rigorously establish the mean-field value ofthis exponent for $d > 4$.

Approaching criticality via the zero dissipation limit in the abelian avalanche model