Alternative proof for the localization of Sinai’s walk

Random Walks in Random Environment (R.W.R.E.) are basic processes in random media. The one dimensional case with nearest neighbor jumps, introduced by Solomon [1975], was first studied by Kesten et al. [1975], Sinai [1982], Golosov [1984], Golosov [1986] and Kesten [1986] all these works show the diversity of the possible behaviors of such walks depending on hypothesis assumed for the environment. At the end of the eighties Deheuvels and Rvsz [1986] and Révész [1989] give the first almost sure behavior of the R.W.R.E. in the recurrent case. Then we have to wait until the middle of the nineties to see new results. An important part of these new results concerns the problem of large deviations first studied by Greven and Hollander [1994] and then by Zeitouni and Gantert [1998], Pisztora and Povel [1999], Zeitouni et al. [1999] and Comets et al. [2000] (see Zeitouni [2001] for a review). In the same period using the stochastic calculus for the recurrent case Shi [1998], Hu and Shi [1998a], Hu and Shi [1998b], Hu [2000a], Hu [2000b] and Hu and Shi [2000] follow the works of Schumacher [1985] and Brox [1986] to give very precise results on the random walk and its local time (see Shi [2001] for an introduction). Moreover recent results on the problem of aging are given in Dembo et al. [2001], on the moderate deviations in Comets and Popov [2003] for the recurrent case, and on the local time in Gantert and Shi [2002] for the transient case. In parallel to all these results a continuous time model has been studied, see for example Schumacher [1985] and Brox [1986], the works of Tanaka [1994], Mathieu [1995], Tanaka [1997], Tanaka and Kawazu [1997], Mathieu [1998] and Taleb [2001]. Since the beginning of the eighties the delicate case of R.W.R.E. in dimension larger than 2 has been studied a lot, see for example Kalikow [1981], Anshelevich et al. [1982], Durrett [1986], Bouchaud et al. [1987], and Bricmont and Kupiainen [1991]. For recent reviews (before 2002) on this topics see the papers of Sznitman [1999] and Zeitouni [2001]. See also Sznitman [2003], Varadhan [2003], Rassoul-Agha [2003] and Comets and Zeitouni [2004].