Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

Abstract

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.

The Divisible Sandpile at Critical Density

“…Related work. The divisible sandpile was introduced in [LP09,LP10] to study the scaling limits of two growth models, rotor aggregation and internal DLA. The divisible sandpile has also been used as a device for proving an exact mean value property for discrete harmonic functions [JLS13, Lemma 2.2].…”

Scaling limits for internal aggregation models with multiple sources

“…Moreover if f ≥ γ is any superharmonic function lying above γ, then f −γ−u is superharmonic on the domain D = {x ∈ Z d |ν(x) = 1} of fully occupied sites, and nonnegative outside D, hence nonnegative everywhere. Thus we have proved the following lemma of [21].…”

“…At each time step, we choose a full site and topple it. As time goes to infinity, provided each full site is eventually toppled, the mass approaches a limiting distribution in which each site has mass ≤ 1; this is proved in [21]. Note that individual topplings do not commute.…”

“…Let σ be a bounded function on R d with compact support. Let D be given by (21), and let D = D ∪ {σ ≥ 1} o . If σ is continuous almost everywhere and satisfies (24), then…”

“…Our shape theorem for multiple point sources, which is deduced from Theorem 1.3 using the main results of [20] and [21], is the following. Theorem 1.4.…”