Logarithmic fluctuations for internal DLA

Details Logarithmic fluctuations for internal DLA Logarithmic fluctuations for internal DLA

2010-10-12

arxiv.org/abs/1010.2483v2

J. Amer. Math. Soc. 25 (2012), 271-301

Mathematics

Probability

38 pages, 5 figures, v2 addresses referee comments. To appear in Journal of the AMS

Scientific paper

Let each of n particles starting at the origin in 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) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{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 C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.

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