Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

We consider a one dimensional random walk in a random environment (RWRE) with wa positive speed lim(n ->infinity) X-n/n = v(alpha) > 0. Gantert and Zeitouni [9] showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P-omega(X-n < xn) with x is an element of (0, v(alpha)) decay approximately like exp {-n(1-1/s)} for a deterministic s > 1. More precisely, they showed that n (gamma)log P-omega(X-n < xn) converges to 0 or 1 depending on whether gamma > 1 - 1/s or gamma < 1 - 1/s. In this paper, we improve on this by showing that n(-1+1/s) log P-omega (X-n < xn) oscillates between 0 and -infinity, almost surely. This had previously been shown only in a very special case of random environments [7].