Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

  • Joe P. Chen

    Colgate University, Hamilton, USA
  • Jonah Kudler-Flam

    The University of Chicago, USA
Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions cover

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Abstract

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpiński gasket (SG) when particles are launched from the corner vertex.

In particular, the abelian sandpile growth problem is exactly solved via a recursive construction of self-similar sandpile tiles. We show that sandpile growth and patterns exhibit a -periodicity as a function of the initial mass. Moreover, the cluster explodes – increments by more than 1 in radius – at periodic intervals, a phenomenon not seen on or trees. We explicitly characterize all the radial jumps, and use the renewal theorem to prove the scaling limit of the cluster radius, which satisfies a power law modulated by log-periodic oscillations. In the course of our proofs we also establish structural identities of the sandpile groups of subgraphs of SG with two different boundary conditions, notably the corresponding identity elements conjectured by Fairchild, Haim, Setra, Strichartz, and Westura.

Our main theorems, in conjunction with recent results of Chen, Huss, Sava-Huss, and Teplyaev, establish SG as a positive example of a state space which exhibits “limit shape universality,” in the sense of Levine and Peres, among the four Laplacian growth models: divisible sandpiles, abelian sandpiles, rotor-router aggregation, and internal diffusion-limited aggregation (IDLA). We conclude the paper with conjectures about radial fluctuations in IDLA on SG, possible extensions of limit shape universality to other state spaces, and related open problems.

Cite this article

Joe P. Chen, Jonah Kudler-Flam, Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), no. 4, pp. 585–664

DOI 10.4171/AIHPD/95