On the concentration of Sinai’s walk

We consider Sinai’s random walk in a random environment. We prove that for an interval of time 1,n]$1,n]$ Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi-totality of this amount of time. Moreover the local time at the point of localization normalized by n$n$ converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time.