  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 an
unoccupied site in $\mathbb{Z}_N \times \mathbb{Z}_{\geq 0}$, which it occupies
forever. This operation defines a Markov chain on subsets of the cylinder. We
first show that a typical subset is rectangular with at most logarithmic
fluctuations. We use this to prove that two Internal DLA chains started from
different 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 the
process started from.
