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

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 \({{\mathbb {Z}}}_N \times {{\mathbb {Z}}}_{\ge 0}\), 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.