Logarithmic fluctuations for internal DLA

Let each of n n particles starting at the origin in Z 2 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 B_r of radius r = n / π r=√n/π . 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 . Br - Clog r ⊂ A(π r^2) ⊂ Br+ Clog r for all sufficiently large r.