## 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

## 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 $1$, \[ \mathbf {B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf {B}_{r+ C\log r} \quad \mbox { for all sufficiently large $r$}. \]## References

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## Bibliographic 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.
- © Copyright 2011 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
- MathSciNet review: 2833484