Scaling Limits for Internal Aggregation Models with Multiple Sources

TL;DR

This paper demonstrates that three different internal aggregation models on a lattice, including stochastic and deterministic types, share the same scaling limit described by a PDE free boundary problem, revealing a universal behavior.

Contribution

It establishes the universal scaling limit of multiple aggregation models as a PDE free boundary problem, linking stochastic and deterministic models to quadrature domains.

Findings

All three models converge to the same PDE-based scaling limit.

The scaling limit is characterized as a quadrature domain.

Results apply to multiple sources and the Diaconis-Fulton smash sum.

Abstract

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.