Localization for Random Walks in Random Environment in Dimension two and Higher

Rodrigo Bazaes

Published 2019-11-15Version 1

In this paper, we introduce the notion of \textit{localization at the boundary} for random walks in i.i.d. and uniformly elliptic random environment, 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 $\omega_{\epsilon}(x,e)=q(e)+\epsilon\xi(x,e)$, where $\{\xi(x)\}_{x\in \mathbb{Z}^{d}}$ is an i.i.d. sequence of random variables, and $\epsilon$ 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.