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

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.**Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni,*Cover times for Brownian motion and random walks in two dimensions*, Ann. of Math. (2)**160**(2004), no. 2, 433–464. MR**2123929**, 10.4007/annals.2004.160.433**3.**Gregory F. Lawler,*On the covering time of a disc by simple random walk in two dimensions*, Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) Progr. Probab., vol. 33, Birkhäuser Boston, Boston, MA, 1993, pp. 189–207. MR**1278083****4.**Gregory F. Lawler,*Intersections of random walks*, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. MR**1117680****5.**Richard Durrett,*Probability: theory and examples*, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR**1609153****6.**Olav Kallenberg,*Foundations of modern probability*, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR**1876169****7.**Pál Révész,*Random walk in random and nonrandom environments*, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. MR**1082348****8.**Thierry Meyre and Wendelin Werner,*Estimation asymptotique du rayon du plus grand disque recouvert par la saucisse de Wiener plane*, Stochastics Stochastics Rep.**48**(1994), no. 1-2, 45–59 (French, with English summary). MR**1786191****9.**P. Erdős and S. J. Taylor,*Some problems concerning the structure of random walk paths*, Acta Math. Acad. Sci. Hungar.**11**(1960), 137–162. (unbound insert) (English, with Russian summary). MR**0121870****10.**Noga Alon and Joel H. Spencer,*The probabilistic method*, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2000. With an appendix on the life and work of Paul Erdős. MR**1885388**

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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:
http://dx.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