Localization at the boundary for conditioned random walks in random environment in dimensions two and higher

Abstract

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on ${\mathbb{Z}}^d$, in dimensions two and higher. If $d=2$ or 3, we prove localization for (almost) all walks. In contrast, for $d\ge 4$, there is a phase transition for environments of the form ${\mathit{\omega }}_{\mathit{\epsilon }}(x,e)=\mathit{\alpha }(e)(1+\mathit{\epsilon }\mathrm{\xi }(x,e))$, where ${\{\mathrm{\xi }(x)\}}_{x\in {\mathbb{Z}}^d}$ is an i.i.d. sequence of random variables, and ε represents the amount of disorder with respect to a simple random walk.