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A note on quenched moderate deviations for Sinai’s random walk in random environment. (English) Zbl 1171.60396
Summary: We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than ${t}^{a}$ ($0) from its initial position, is $exp\left\{-\mathrm{Const}·{t}^{a}/\left[\left(1-a\right)lnt\right]\left(1+o\left(1\right)\right)\right\}$.
##### MSC:
 60K37 Processes in random environments 60F10 Large deviations 82B41 Random walks, random surfaces, lattice animals, etc. (statistical mechanics) 82B44 Disordered systems (equilibrium statistical mechanics)