Convergence of the random Abelian sandpile

TL;DR

This paper proves that Abelian sandpiles with random initial states almost surely converge to unique scaling limits, using stochastic homogenization techniques, and extends results to divisible sandpiles with explicit scaling limits.

Contribution

It introduces a new proof of convergence for random Abelian sandpiles and identifies the scaling limit of divisible sandpiles, advancing understanding of their asymptotic behavior.

Findings

Almost sure convergence to unique scaling limits

Identification of the divisible sandpile's scaling limit

Quantitative proof of stabilizability of Abelian sandpiles

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.