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.
Acknowledgments
I am grateful to Charles K. Smart for suggesting the program in [2], patiently providing essential advice throughout this project and carefully reviewing a previous draft of this paper. I am also grateful to Steven P. Lalley for useful conversations, encouragement and first introducing me to this problem. I thank Lionel Levine for generous, detailed comments on a previous draft and for helpful discussions. I also acknowledge Khalid Bou-Rabee, Nawaf Bou-Rabee, Gregory Lawler and Micol Tresoldi for inspiring conversations. The anonymous referees also provided detailed feedback, which led to a much improved exposition.
Citation
Ahmed Bou-Rabee. "Convergence of the random Abelian sandpile." Ann. Probab. 49 (6) 3168 - 3196, November 2021. https://doi.org/10.1214/21-AOP1528
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