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

2011/01/07 by Járai, Antal A.
#60K35 (Primary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1101.1437

Abstract

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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