Logarithmic Fluctuations for Internal DLA

Let each of n particles starting at the origin in Z 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 Br of radius r = √ n/π. 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 with probability 1, Br−C log r ⊂ A(πr) ⊂ Br+C log r for all sufficiently large r. ∗Partially supported by NSF grant DMS-1069225 †Supported by an NSF Postdoctoral Research Fellowship. ‡Partially supported by NSF grant DMS-0645585. 2010 Mathematics Subject Classification: 60G50, 60K35, 82C24. 1 ar X iv :1 01 0. 24 83 v2 [ m at h. PR ] 9 J ul 2 01 1