Containing Internal Diffusion Limited Aggregation

TL;DR

This paper establishes a limit-shape theorem for Internal Diffusion Limited Aggregation (IDLA) on supercritical percolation clusters, introducing a new, robust technique for proving upper bounds that relies only on a good lower bound.

Contribution

It provides the first upper bound for IDLA shape on percolation clusters, using a novel method that avoids harmonic measure estimates, applicable to random environments.

Findings

Proves a limit-shape theorem for IDLA on percolation clusters.

Introduces a new technique for upper bounds based on lower bounds.

Demonstrates robustness of the method in random environments.

Abstract

Internal Diffusion Limited Aggregation (IDLA) is a model that describes the growth of a random aggregate of particles from the inside out. Shellef proved that IDLA processes on supercritical percolation clusters of integer-lattices fill Euclidean balls, with high probability. In this article, we complete the picture and prove a limit-shape theorem for IDLA on such percolation clusters, by providing the corresponding upper bound. The technique to prove upper bounds is new and robust: it only requires the existence of a "good" lower bound. Specifically, this way of proving upper bounds on IDLA clusters is more suitable for random environments than previous ways, since it does not harness harmonic measure estimates.