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

Abstract: We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski 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 $(2\cdot 3<sup>n)$-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 $\mathbb{Z}<sup>d$ 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.