Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment

2007/08/04 by Peterson, Jonathon
#60F05 #60K37 #82C41 #82D30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0708.0649

Abstract

We consider a nearest-neighbor, one-dimensional random walk \X_n\_{n≥ 0} in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an s∈(1,2) such that the annealed law of n^-1/s (X_n - n v_P) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences t_k and t_k' depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0,1).

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