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We consider a continuous height version of the Abelian sandpile model withsmall amount of bulk dissipation gamma > 0 on each toppling, in dimensions d =2, 3. In the limit gamma -> 0, we give a power law upper bound, based oncoupling, on the rate at which the stationary measure converges to the discretecritical sandpile measure. The proofs are based on a coding of the stationarymeasure by weighted spanning trees, and an analysis of the latter via Wilson'salgorithm. In the course of the proof, we prove an estimate on coupling ageometrically killed loop-erased random walk to an unkilled loop-erased randomwalk.

Rate of convergence estimates for the zero dissipation limit in Abelian sandpiles