The Annals of Probability

A weakness in strong localization for Sinai’s walk

Zhan Shi and Olivier Zindy

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Abstract

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.

Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 1118-1140.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1178804324

Digital Object Identifier
doi:10.1214/009117906000000863

Mathematical Reviews number (MathSciNet)
MR2319717

Zentralblatt MATH identifier
1117.60091

Subjects
Primary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks 60J55: Local time and additive functionals

Keywords
Random walk in random environment favorite site local time occupation time

Citation

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


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