Open Access
May 2007 A weakness in strong localization for Sinai’s walk
Zhan Shi, Olivier Zindy
Ann. Probab. 35(3): 1118-1140 (May 2007). DOI: 10.1214/009117906000000863


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.


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Zhan Shi. Olivier Zindy. "A weakness in strong localization for Sinai’s walk." Ann. Probab. 35 (3) 1118 - 1140, May 2007.


Published: May 2007
First available in Project Euclid: 10 May 2007

zbMATH: 1117.60091
MathSciNet: MR2319717
Digital Object Identifier: 10.1214/009117906000000863

Primary: 60G50 , 60J55 , 60K37

Keywords: Favorite site , Local time , occupation time , Random walk in random environment

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • May 2007
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