Internal aggregation models with multiple sources and obstacle problems on Sierpiński gaskets

Abstract

We consider the doubly infinite Sierpiński gasket graph ${\mathsf{SG}}_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs ${\mathsf{SG}}_n=2^{-n}{\mathsf{SG}}_0$, for every $n\in \mathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration ${\sigma }_n$ of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by $\mathsf{SG}=\text{cl}({\bigcup }_{n=0}^{\infty }{\mathsf{SG}}_n)$ the infinite Sierpiński gasket, which is a closed subset of ${\mathbb{R}}^2$, for which ${\mathsf{SG}}_n$ represents the level-$n$ approximating graph, and we consider a continuous function $\sigma :\mathsf{SG}\to \mathbb{N}$. For $\sigma$ we solve the obstacle problem, and we describe the noncoincidence set $D\subset \mathsf{SG}$ as the solution of a free boundary problem on the fractal $\mathsf{SG}$. If the discrete particle configurations ${\sigma }_n$ on the approximating graphs ${\mathsf{SG}}_n$ converge pointwise to the continuous function $\sigma$ on the limit set $\mathsf{SG}$, we prove that, as $n\to \infty$, the scaling limits of the three aforementioned models on ${\mathsf{SG}}_n$ starting with initial particle configuration ${\sigma }_n$ converge to the deterministic solution $D$ of the free boundary problem on the limit set $\mathsf{SG}\subset {\mathbb{R}}^2$. For $D$ we also investigate boundary regularity properties.