3 research outputs found
Approaching criticality via the zero dissipation limit in the abelian avalanche model
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 .Comment: 46 pages, substantially revised 4th version, title has been changed.
The main new material is Section 6 on toppling probabilities and the toppling
probability exponen