Oscillation of quenched slowdown asymptotics of random walks in random environment in Z

Sung Won Ahn, Purdue University


We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, we improve on this by showing that n -1+1/s log P ω(Xn < xn) oscillates between 0 and -∞ , almost surely.




Peterson, Purdue University.

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