Convergence of the random Abelian sandpile

Abstract

We prove that Abelian sandpiles with random initial states converge almost surely to unique scaling limits. The proof follows the Armstrong-Smart program for stochastic homogenization of uniformly elliptic equations. Using simple random walk estimates, we prove an analogous result for the divisible sandpile and identify its scaling limit as exactly that of the averaged divisible sandpile. As a corollary, this gives a new quantitative proof of known results on the stabilizability of Abelian sandpiles.

31 pages, 5 figures; minor revisions addressing comments from anonymous referees

References in corpus (1)

• Stabilizability and percolation in the infinite volume sandpile model

Cited by in corpus (1)

• Dynamic dimensional reduction in the Abelian sandpile