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

Abstract

We introduce the notion of \emph{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. Informally, this means that the walk spends a non-trivial amount of time at some point $x\in \mathbb{Z}^{d}$ with $|x|_{1}=n$ at time $n$, for $n$ large enough. In dimensions two and three, we prove localization for (almost) all walks. In contrast, for $d\geq 4$ there is a phase-transition for environments of the form $\omega_{\varepsilon}(x,e)=\alpha(e)(1+\varepsilon\xi(x,e))$, where $\{\xi(x)\}_{x\in \mathbb{Z}^{d}}$ is an i.i.d. sequence of random variables, and $\varepsilon$ represents the amount of disorder with respect to a simple random walk. The proofs involve a criterion that connects localization with the equality or difference between the quenched and annealed rate functions at the boundary.