A note on fluctuations for internal diffusion limited aggregation

TL;DR

This paper improves the fluctuation bounds for the shape of internal diffusion limited aggregation clusters in dimensions three and higher, introducing a new coupled process called the flashing process to achieve tighter estimates.

Contribution

It introduces the flashing process, a new coupled growth model, to better control fluctuations in internal DLA clusters in higher dimensions.

Findings

Fluctuations are bounded by n^{1/(d+1)} in dimensions ≥ 3.

The flashing process simplifies fluctuation control.

Coupling with the flashing process yields improved bounds.

Abstract

We consider a cluster growth model on Z^d, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. Also, when dimension is 2 or more, and when the cluster has volume $n^d$, it is known that fluctuations of the radius are at most of order $n^{1/3}$. We improve this estimate to $n^{1/(d+1)}$, in dimension 3 or more. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachere in our Appendix) on the expected time spent by a random walk inside an annulus.