Quenched limits for transient, zero speed one-dimensional random walk in random environment

Abstract

We consider a nearest-neighbor, one dimensional random walk $\{X_n{\}}_{n\ge 0}$ in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that $X_n$ is of order $n^s$ for some $s<1$. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences $\{n_k\}$ and $\{x_k\}$ depending on the environment only, such that $X_{n_k}-x_k=o(\log n_k{)}^2$ (a localized regime). On the other hand, there exist sequences $\{t_m\}$ and $\{s_m\}$ depending on the environment only, such that $\log s_m/\log t_m\to s<1$ and $P_{\omega }(X_{t_m}/s_m\le x)\to 1/2$ for all $x>0$ and $\to 0$ for $x\le 0$ (a spread out regime).

Published in at http://dx.doi.org/10.1214/08-AOP399 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Limit laws for transient random walks in random environment on $\z$
• Stable fluctuations for ballistic random walks in random environment on Z

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