Fluctuations for internal DLA on the Comb

TL;DR

This paper investigates the shape fluctuations of internal DLA on the two-dimensional comb lattice, revealing Gaussian fluctuations similar to those in one-dimensional lattices, and provides insights into the asymptotic shape of the cluster.

Contribution

It demonstrates that fluctuations of internal DLA on the comb lattice are Gaussian, extending understanding of growth models on non-Euclidean structures.

Findings

Fluctuations are Gaussian on the comb lattice.

The asymptotic shape aligns with previous bounds.

Provides new insights into DLA behavior on tree-like structures.

Abstract

We study internal diffusion limited aggregation (DLA) on the two dimensional comb lattice. The comb lattice is a spanning tree of the euclidean lattice, and internal DLA is a random growth model, where simple random walks, starting one at a time at the origin of the comb, stop when reaching the first unoccupied site. An asymptotic shape is suggested by a lower bound of Huss and Sava. We show that fluctuations with respect to this shape are gaussian as in the one-dimensional lattice.