Diamond Aggregation

TL;DR

This paper investigates how the shape of aggregates in a diffusion-limited growth model depends on the underlying random walk law, revealing conditions for diamond-shaped limits and fluctuation bounds.

Contribution

It introduces a family of biased walks in Z^2 that produce diamond-shaped limits and demonstrates logarithmic fluctuation bounds, contrasting with the disk shape of simple random walks.

Findings

Diamond-shaped limiting forms for certain walks

Logarithmic fluctuation bounds for biased walks

Contrast with power-law fluctuations in simple random walk

Abstract

Internal diffusion-limited aggregation is a growth model based on random walk in Z^d. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in Z^2 for which the limiting shape is a diamond. Certain of these walks -- those with a directional bias toward the origin -- have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.