Discrete Balayage and Boundary Sandpile

TL;DR

This paper introduces a boundary sandpile model on lattices, analyzing its properties, regularity, and convergence, and connects it to free boundary problems and classical sandpile limits.

Contribution

It develops a new lattice growth model with potential-theoretic redistribution, providing regularity results and convergence analysis, and links to classical sandpile boundary behavior.

Findings

The model admits a canonical super-solution representation.

The scaled odometer function is uniformly Lipschitz continuous.

The boundary of the scaling limit of classical Abelian sandpile is Lipschitz.

Abstract

We introduce a new lattice growth model, which we call boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on $\mathbb{Z}^d$ ($d\geq 2$) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile. As a direct application of some of the methods developed in this paper, combined with earlier results on classical Abelian sandpile, we show that the boundary of the scaling limit of Abelian sandpile is locally a Lipschitz graph.