Logarithmic fluctuations for internal DLA

TL;DR

This paper proves that the shape of the internal DLA cluster in Z^2 closely approximates a disk with a discrepancy that grows logarithmically with the radius, refining previous shape estimates.

Contribution

It establishes a logarithmic bound on the discrepancy between the internal DLA cluster and the ideal disk shape in Z^2.

Findings

Discrepancy between the cluster and the disk is at most logarithmic in radius

The shape of the cluster is tightly bounded within logarithmic deviations

Results hold with probability one for large radii

Abstract

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/\pi}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.