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 be the radius of the largest disk covered after steps of a simple random walk. We prove that almost surely

**1.**J. Hough,*Asymptotic results for zeros of diffusing Gaussian analytic functions*, Ph.D. dissertation, University of California, Berkeley (2006).**2.**A. Dembo, Y. Peres, J. Rosen and O. Zeitouni,*Cover Times for Brownian motion and random walks in two dimensions*, Annals Math. (2) 160 (2004), 433-464. MR**2123929 (2005k:60261)****3.**G. Lawler,*On the covering time of a disk by simple random walk in two dimensions*, In Seminar on Stochastic Processes 1992, 189-208. Birkhauser (1993). MR**1278083 (95c:60064)****4.**G. Lawler,*Intersections of random walks*. Birkhauser (1991).MR**1117680 (92f:60122)****5.**R. Durrett,*Probability: Theory and Examples*. Duxbury (1996).MR**1609153 (98m:60001)****6.**O. Kallenberg,*Foundations of Modern Probability*. Springer (2002). MR**1876169 (2002m:60002)****7.**P. Révész,*Random Walk in Random and Non-random Environments*. World Scientific (1990). MR**1082348 (92c:60096)****8.**T. Meyre, W. Werner,*Estimation asymptotique du rayon du plus grand disque recouvert par la saucisse de Wiener plane*, Stochastics 48 (1994), 45-59. MR**1786191 (2001e:60171)****9.**P. Erdös, S.J. Taylor,*Some problems concerning the structure of random walk paths*, Acta Math. Acad. Sci. Hungar. 11 (1960), 137-162. MR**0121870 (22:12599)****10.**N. Alon and J. Spencer,*The Probabilistic Method*, Second Edition, Wiley, 2000. MR**1885388 (2003f:60003)**

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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