Internal DLA on cylinder graphs: fluctuations and mixing

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 $\mathcal{O}( N\sqrt{τ_N} (\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $τ_N$ 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.

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