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.
- J. Hough, Asymptotic results for zeros of diffusing Gaussian analytic functions, Ph.D. dissertation, University of California, Berkeley (2006).
- 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, DOI https://doi.org/10.4007/annals.2004.160.433
- 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
- Gregory F. Lawler, Intersections of random walks, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1117680
- Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
- Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169
- Pál Révész, Random walk in random and nonrandom environments, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. MR 1082348
- 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, DOI https://doi.org/10.1080/17442509408833897
- 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 121870, DOI https://doi.org/10.1007/BF02020631
- 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
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
