The Annals of Probability

A weakness in strong localization for Sinai’s walk

Zhan Shi and Olivier Zindy

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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

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

First available in Project Euclid: 10 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random walk in random environment favorite site local time occupation time


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.

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  • Andreoletti, P. (2005). Almost sure estimates for the concentration neighborhood of Sinai's walk. Available at
  • Bertoin, J. (1993). Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 17--35.
  • Dembo, A., Gantert, N., Peres, Y. and Shi, Z. (2005). Valleys and the maximum local time for random walk in random environment. Available at
  • Erdős, P. and Révész, P. (1984). On the favourite points of a random walk. In Mathematical Structures---Computational Mathematics---Mathematical Modelling 2 152--157. Bulgaria Academy of Sciences, Sofia.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • Golosov, A. O. (1984). Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 491--506.
  • den Hollander, F. (2000). Large Deviations. Amer. Math. Soc., Providence, RI.
  • Révész, P. (1990). Random Walk in Random and Non-Random Environments. World Scientific, Teaneck, NJ.
  • Shi, Z. (1998). A local time curiosity in random environment. Stochastic Process. Appl. 76 231--250.
  • Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York.
  • Sinai, Ya. G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 256--268.
  • Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1--31.
  • Zeitouni, O. (2004). Random walks in random environment. XXXI Summer School in Probability, St Flour (2001). Lecture Notes in Math. 1837 193--312. Springer, Berlin.