We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on , in dimensions two and higher. If or 3, we prove localization for (almost) all walks. In contrast, for , there is a phase transition for environments of the form , where is an i.i.d. sequence of random variables, and ε represents the amount of disorder with respect to a simple random walk.
The author has been supported by ANID-PFCHA/Doctorado Nacional/2018-21180873.
The author thanks his advisor Alejandro F. Ramírez and an anonymous referee for valuable comments and suggestions about the paper. Also, the author is grateful to Gregorio Moreno for a couple of discussions and bibliographical remarks.
"Localization at the boundary for conditioned random walks in random environment in dimensions two and higher." Electron. Commun. Probab. 26 1 - 13, 2021. https://doi.org/10.1214/21-ECP426