Internal DLA on cylinder graphs: fluctuations and mixing

TL;DR

This paper studies the internal DLA process on cylinder graphs, showing it forgets initial configurations in a time depending on the base graph's size and mixing time, with new bounds based on mixing properties.

Contribution

Introduces a maximal fluctuation bound for IDLA on cylinder graphs using mixing properties and the Abelian property, extending previous coupling methods.

Findings

IDLA on cylinder graphs forgets initial profiles in a specific time scale.

New fluctuation bounds depend only on base graph mixing times.

Results apply to large base graphs with known mixing properties.

Abstract

We use coupling ideas introduced in \cite{levine2018long} to show that an IDLA process on a cylinder graph $G\times \mathbb{Z}$ forgets a typical initial profile in $\mathcal{O}( N\sqrt{\tau_N} (\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $\tau_N$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb{Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.