An LIL for cover times of disks by planar random walk and Wiener sausage
HTML articles powered by AMS MathViewer
- by J. Ben Hough and Yuval Peres PDF
- Trans. Amer. Math. Soc. 359 (2007), 4653-4668
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.References
- 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 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, DOI 10.1007/978-1-4757-4015-8
- Pál Révész, Random walk in random and nonrandom environments, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. MR 1082348, DOI 10.1142/1107
- 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 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 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, DOI 10.1002/0471722154
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
- © Copyright 2007 by 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
- MathSciNet review: 2320645