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
Full-text PDF Free Access
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 \[ \limsup _{n \rightarrow \infty }(\log R_n)^2/(\log n \log _3 n) = 1/4,\] 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.
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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
Article copyright:
© Copyright 2007
by J. Ben Hough and Yuval Peres