Logarithmic Fluctuations for Internal DLA

Abstract

Let each of [superscript n] particles starting at the origin in Z[superscript 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 [superscript n] occupied sites is (with high probability) close to a disk B [subscript r] of radius r=√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 [superscript C] such that with probability [superscript 1], B [subscript r - C log r] C A(pi r[superscript 2]) C B [subscript r+ C log r] for all sufficiently large r.