## An LIL for cover times of disks by planar random walk and Wiener sausage

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- by J. Ben Hough and Yuval Peres PDF
- Trans. Amer. Math. Soc.
**359**(2007), 4653-4668

## Abstract:

Let $R_n$ be the radius of the largest disk covered after $n$ steps of a simple random walk. We prove that almost surely \[ \limsup _{n \rightarrow \infty }(\log R_n)^2/(\log n \log _3 n) = 1/4,\] where $\log _3$ denotes 3 iterations of the $\log$ function. This is motivated by a question of Erdős and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.## References

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## Additional Information

**J. Ben Hough**- Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
- Address at time of publication: HBK Capital Management, 350 Park Avenue, Fl 20, New York, New York 10022
- Email: jbhough@math.berkeley.edu
**Yuval Peres**- Affiliation: Departments of Statistics and Mathematics, University of California Berkeley, Berkeley, California 94720
- MR Author ID: 137920
- Email: peres@stat.berkeley.edu
- Received by editor(s): September 18, 2004
- Received by editor(s) in revised form: January 5, 2005
- Published electronically: May 1, 2007
- Additional Notes: The authors gratefully acknowledge the financial support from NSF grants $\#$DMS-0104073 and $\#$DMS-0244479
- © Copyright 2007 by J. Ben Hough and Yuval Peres
- Journal: Trans. Amer. Math. Soc.
**359**(2007), 4653-4668 - MSC (2000): Primary 60F15
- DOI: https://doi.org/10.1090/S0002-9947-07-03966-9
- MathSciNet review: 2320645