Abstract. We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1): [431][432][433][434][435][436][437][438][439][440][441][442][443][444][445][446][447][448][449][450] 2008). For the shape consisting of n = ω d r d sites, where ω d is the volume of the unit ball in R d , we show that the inradius of the set of occupied sites is at least r − O(log r), while the outradius is at most r + O(r α ) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr 2 particles, we show that the inradius is at least r/ √ 3, and the outradius is at most (r + o(r))/ √ 2. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.
Let each of n particles starting at the origin in Z 2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B r of radius r = n/π. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that with probability 1, B r−C log r ⊂ A(πr 2 ) ⊂ B r+C log r for all sufficiently large r.
We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z d . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h ≤ 2d − 2, the diameter of the set of sites that topple has order n 1/d . This was previously known only for h < d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification.We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d − 1. On the other hand, we show that if the background height 2d − 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).
We study the scaling limits of three different aggregation models on Z d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the DiaconisFulton smash sum of domains.
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