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We introduce the notion of \emph{localization at the boundary} forconditioned random walks in i.i.d. and uniformly elliptic random environment on$\mathbb{Z}^d$, in dimensions two and higher. Informally, this means that thewalk 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 andthree, 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 simplerandom walk. The proofs involve a criterion that connects localization with theequality or difference between the quenched and annealed rate functions at theboundary.

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