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

TL;DR

This paper investigates how quickly the stationary distribution of a dissipative Abelian sandpile model converges to the critical measure as dissipation vanishes, using coupling and spanning tree techniques in dimensions 2 and 3.

Contribution

It provides a power law upper bound on the convergence rate for the zero dissipation limit in Abelian sandpiles, employing coupling and Wilson's algorithm.

Findings

Established a power law upper bound on convergence rate

Developed a coupling method for loop-erased random walks

Connected stationary measures to weighted spanning trees

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.