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An LIL for cover times of disks by planar random walk and Wiener sausage

J. Ben Hough·Yuval Peres
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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 \to \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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J. Ben Hough, Yuval Peres. 2005-01-24. An LIL for cover times of disks by planar random walk and Wiener sausage. https://arxiv.org/abs/math/0409239

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