From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

TL;DR

This paper establishes logarithmic and polynomial bounds on fluctuations of internal DLA cluster growth in higher dimensions, introducing a new flashing process to facilitate analysis.

Contribution

It introduces the flashing process, a new growth model, to effectively control and analyze fluctuations of internal DLA clusters in multiple dimensions.

Findings

Fluctuations are at most a power of the logarithm of the radius in dimensions ≥ 2.

The flashing process provides a tractable way to bound cluster fluctuations.

The approach adapts existing methods and incorporates sharp estimates on random walk behavior.

Abstract

We consider a cluster growth model on the d-dimensional lattice, 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. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d larger than or equal to 2. 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 Blach\`ere in our Appendix) on the expected time spent by a random walk inside an annulus.