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
DOI:
https://doi.org/10.1090/S0894-0347-2011-00716-9
Published electronically:
August 15, 2011
MathSciNet review:
2833484
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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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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
MR Author ID:
654666
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
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