A weakness in strong localization for Sinai's walk

Details A weakness in strong localization for Sinai's walk A weakness in strong localization for Sinai's walk

2006-06-15

arxiv.org/abs/math/0606376v2

Annals of Probability 2007, Vol. 35, No. 3, 1118-1140

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000863 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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 Erd\H{o}s and R\'{e}v\'{e}sz [Mathematical Structures--Computational Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated for the usual homogeneous random walk.

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