Laplacian growth, sandpiles and scaling limits

TL;DR

This paper surveys recent progress in discrete Laplacian growth models like sandpiles and internal DLA, discusses their scaling limits, and presents a new result on rotor aggregation fluctuations.

Contribution

It provides a comprehensive review of discrete Laplacian growth models and introduces a new bound on fluctuations in rotor aggregation in Z^d.

Findings

Rotor aggregation in Z^d has O(log r) fluctuations.

Models illustrate tools of discrete potential theory.

Open questions remain on fluctuation bounds.

Abstract

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z^d as the mesh size goes to zero. These models provide a window into the tools of discrete potential theory: harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in Z^d has O(log r) fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are O(1).