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

TL;DR

This paper investigates the time scale over which Internal DLA on a cylinder forgets its initial configuration, establishing bounds on the number of particles needed for the process to lose initial profile information.

Contribution

It provides the first quantitative bounds on the mixing time of Internal DLA on a cylinder, showing how many particles are needed to forget the initial state.

Findings

Typical Internal DLA subsets are rectangular with logarithmic fluctuations.

Coupling of two chains from different initial states requires order N^2 log N particles.

At least order N^2 particles are needed to forget initial configurations.

Abstract

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.