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

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

2007-04-13

arxiv.org/abs/0704.1778v2

Annals of Probability 2009, Vol. 37, No. 1, 143-188

Mathematics

Probability

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

Scientific paper

10.1214/08-AOP399

We consider a nearest-neighbor, one dimensional random walk $\{X_n\}_{n\geq0}$ 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 $P_{\omega}(X_{t_m}/s_m\leq x)\to1/2$ for all $x>0$ and $\to0$ for $x\leq0$ (a spread out regime).

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