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Transactions of the American Mathematical Society

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


Authors: 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
Published electronically: May 1, 2007
MathSciNet review: 2320645
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Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \limsup_{n \rightarrow \infty}(\log R_n)^2/(\log n \log_3 n) = 1/4,$

where $ \log_3$ denotes 3 iterations of the $ \log$ function. This is motivated by a question of Erdos and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.


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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
Email: peres@stat.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-07-03966-9
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
Article copyright: © Copyright 2007 by J. Ben Hough and Yuval Peres

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