We consider Sinai's random walk in random environment. We prove that for an interval of time [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 $n$ converges in probability to a well defined random variable of the environment.
Constants of concentration for a simple recurrent random walk on random environment
A limit result for a system of particles in random environment
Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime
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