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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

Logarithmic fluctuations for internal DLA


Authors: David Jerison, Lionel Levine and Scott Sheffield
Journal: J. Amer. Math. Soc. 25 (2012), 271-301
MSC (2010): Primary 60G50, 60K35, 82C24
Published electronically: August 15, 2011
MathSciNet review: 2833484
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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$,

$\displaystyle \mathbf{B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf{B}_{r+ C\log r}$   $\displaystyle \mbox{ for all sufficiently large $r$}$$\displaystyle . $


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Additional Information

David Jerison
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: jerison@math.mit.edu

Lionel Levine
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: levine@math.mit.edu

Scott Sheffield
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: sheffield@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00716-9
PII: S 0894-0347(2011)00716-9
Received by editor(s): December 3, 2010
Received by editor(s) in revised form: July 8, 2011
Published electronically: August 15, 2011
Additional Notes: This work was supported by NSF grants DMS-1069225 and DMS-0645585 and an NSF Postdoctoral Research Fellowship.
Article copyright: © Copyright 2011 David Jerison, Lionel Levine, and Scott Sheffield