Internal DLA on cylinder graphs: fluctuations and mixing

Abstract

We use coupling ideas introduced in [13] 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.