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 $d\ge 4$ there is a phase-transition for environments of the form ${\omega }_{\epsilon }(x,e)=\alpha (e)(1+\epsilon \xi (x,e))$, 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.

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