Gaussian fluctuations for Internal DLA on cylinders

TL;DR

This paper extends the understanding of internal DLA fluctuations on cylinders, showing they converge to a Gaussian free field for a broad class of base graphs, and improves bounds on cluster fluctuations.

Contribution

It generalizes fluctuation results to any vertex-transitive base graph satisfying eigenvalue convergence, and provides an improved fluctuation bound and shape theorem.

Findings

Fluctuations on cylinders with general base graphs converge to GFF.

Established an improved bound on maximal cluster fluctuations.

Proved a shape theorem for IDLA on these cylinders.

Abstract

Internal DLA is a discrete random growth model describing growing clusters of particles. Its limiting shape and fluctuations are well understood when the underlying graph is the $d$-dimensional lattice or the cylinder $\mathbb{Z}_N \times \mathbb{Z}$. In the latter geometry, the average fluctuations of IDLA have been shown to converge to the GFF. In this note we generalise this result by showing that, for any vertex-transitive base graph $V_N$ satisfying an eigenvalue convergence condition, the average fluctuations of IDLA on the cylinder $V_N \times \mathbb{Z}$ are given by a GFF. On the way, we present an improved bound on the clusters' maximal fluctuations, which is of independent interest and which implies a shape theorem for IDLA on $V_N \times \mathbb{Z}$ for any vertex-transitive base graph $V_N$.