The Annals of Probability
- Ann. Probab.
- Volume 35, Number 3 (2007), 1118-1140.
A weakness in strong localization for Sinai’s walk
Sinai’s walk is a recurrent one-dimensional nearest-neighbor random walk in random environment. It is known for a phenomenon of strong localization, namely, the walk spends almost all time at or near the bottom of deep valleys of the potential. Our main result shows a weakness of this localization phenomenon: with probability one, the zones where the walk stays for the most time can be far away from the sites where the walk spends the most time. In particular, this gives a negative answer to a problem of Erdős and Révész [Mathematical Structures—Computational Mathematics—Mathematical Modelling 2 (1984) 152–157], originally formulated for the usual homogeneous random walk.
Ann. Probab., Volume 35, Number 3 (2007), 1118-1140.
First available in Project Euclid: 10 May 2007
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Shi, Zhan; Zindy, Olivier. A weakness in strong localization for Sinai’s walk. Ann. Probab. 35 (2007), no. 3, 1118--1140. doi:10.1214/009117906000000863. https://projecteuclid.org/euclid.aop/1178804324