Logarithmic Fluctuations for Internal DLA

• Main Authors: Jerison, David S. (Contributor), Levine, Lionel (Contributor), Sheffield, Scott Roger (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: American Mathematical Society (AMS), 2012-07-03T12:39:01Z.
• Subjects: Article
• Online Access: Get fulltext

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.
National Science Foundation (U.S.) (grant DMS-1069225)
National Science Foundation (U.S.) (grant DMS-0645585)
National Science Foundation (U.S.) (Postdoctoral Research Fellowship)