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Internal DLA is a discrete model of a moving interface. On the cylinder graph$\mathbb{Z}_N \times \mathbb{Z}$, a particle starts uniformly on $\mathbb{Z}_N\times \{0\}$ and performs simple random walk on the cylinder until reaching anunoccupied site in $\mathbb{Z}_N \times \mathbb{Z}_{\geq 0}$, which it occupiesforever. This operation defines a Markov chain on subsets of the cylinder. Wefirst show that a typical subset is rectangular with at most logarithmicfluctuations. We use this to prove that two Internal DLA chains started fromdifferent typical subsets can be coupled with high probability by adding order$N^2 \log N$ particles. For a lower bound, we show that at least order $N^2$particles are required to forget which of two independent typical subsets theprocess started from.

How long does it take for Internal DLA to forget its initial profile?