A weakness in strong localization for Sinai's walk
Résumé
Sinai's walk is a recurrent one-dimensional nearest-neighbour 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 os and Révész \cite{erdos-revesz}, originally formulated for the usual homogeneous random walk.