Logarithmic fluctuations for internal DLA

Jerison, David; Levine, Lionel; Sheffield, Scott

doi:10.1090/S0894-0347-2011-00716-9

Logarithmic fluctuations for internal DLA
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by David Jerison, Lionel Levine and Scott Sheffield

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

DOI: https://doi.org/10.1090/S0894-0347-2011-00716-9

Published electronically: August 15, 2011

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Abstract:

Let each of $n$ particles starting at the origin in $\mathbb Z^2$ 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 $\mathbf {B}_r$ 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 with probability $1$, \[ \mathbf {B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf {B}_{r+ C\log r} \quad \mbox { for all sufficiently large $r$}. \]

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