Logarithmic fluctuations for internal DLA

Abstract

Let each of n n particles starting at the origin in Z 2 \mathbb 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 ) A(n) of n n occupied sites is (with high probability) close to a disk B r \mathbf {B}_r of radius r = n / π r=\sqrt {n/\pi } . We show that the discrepancy between A ( n ) A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant  C C such that with probability 1 1 , \[ B r − C log ⁡ r ⊂ A ( π r 2 ) ⊂ B r + C log ⁡ r  for all sufficiently large  r . \mathbf {B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf {B}_{r+ C\log r} \quad \mbox { for all sufficiently large $r$}. \]