Ju n 20 09 Zero dissipation limit in the Abelian sandpile model

We study the abelian avalanche model, an analogue of the abelian sandpile model with continuous heights, which allows for arbitrary small values of dissipation. We prove that for non-zero dissipation, the infinite volume limit of the stationary measures of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. Moreover we obtain 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 measures and for the dynamics. Key-words: Abelian avalanche model, burning algorithm, weighted spanning trees, Wilson’s algorithm, zero-dissipation limit, self-organized criticality.