Ricerc@Sapienza

Internal DLA is a discrete model of a moving interface. On the cylinder graph ZN× Z, a particle starts uniformly on ZN× 0 and performs simple random walk on the cylinder until reaching an unoccupied site in ZN× Z≥ 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 N2log N particles.

We use coupling ideas introduced in [13] to show that an IDLA process on a cylinder graph G × Z forgets a typical initial profile in O(N√τN (logN)2) steps for large N, where N is the size of the base graph G, and τN 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 × Z which only relies on the mixing properties of the base graph G and the Abelian property.