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):431–450, 2008). For the shape consisting of \(n=\omega_d r^d\) sites, where ω d is the volume of the unit ball in \(\mathbb{R}^d\), we show that the inradius of the set of occupied sites is at least r − O(logr), 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 \(r/\sqrt{3}\), and the outradius is at most \((r+o(r))/\sqrt{2}\). This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.

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Correspondence to Lionel Levine.

Additional information

Lionel Levine is supported by an NSF Graduate Research Fellowship, and NSF grant DMS-0605166.

Yuval Peres is partially supported by NSF grant DMS-0605166.

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Levine, L., Peres, Y. Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile. Potential Anal 30, 1 (2009). https://doi.org/10.1007/s11118-008-9104-6

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Keywords

  • Abelian sandpile
  • Asymptotic shape
  • Discrete Laplacian
  • Divisible sandpile
  • Growth model
  • Internal diffusion limited aggregation
  • Rotor-router model

Mathematics Subject Classifications (2000)

  • Primary 60G50
  • Secondary 35R35