Convergence of the random Abelian sandpile
arXiv:1909.07849 · doi:10.1214/21-AOP1528
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.
arXiv comment31 pages, 5 figures; minor revisions addressing comments from anonymous referees
References in corpus (10)
- Chip-Firing and Rotor-Routing on Directed Graphs
- Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
- Scaling Limits for Internal Aggregation Models with Multiple Sources
- Algebraic Aspects of Abelian Sandpile Models
- Growth Rates and Explosions in Sandpiles
- Convergence of the Abelian sandpile
- Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity
- Stabilizability and percolation in the infinite volume sandpile model
- The divisible sandpile at critical density
- Sandpiles on the square lattice