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

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

2011-01-07

arxiv.org/abs/1101.1437v2

Mathematics

Probability

28 pages; Section 2 has been substantially expanded, giving more background

Scientific paper

We consider a continuous height version of the Abelian sandpile model with small 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 on coupling, on the rate at which the stationary measure converges to the discrete critical sandpile measure. The proofs are based on a coding of the stationary measure by weighted spanning trees, and an analysis of the latter via Wilson's algorithm. In the course of the proof, we prove an estimate on coupling a geometrically killed loop-erased random walk to an unkilled loop-erased random walk.

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