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

TL;DR

This paper studies how conditioned random walks in random environments tend to localize at boundary points in dimensions two and higher, revealing dimension-dependent behaviors and phase transitions influenced by environmental disorder.

Contribution

It introduces the concept of boundary localization for conditioned random walks and establishes results on localization phenomena across different dimensions, including phase transitions in higher dimensions.

Findings

Localization occurs in dimensions two and three for almost all walks.

A phase transition in localization behavior is identified for dimensions four and higher.

The study links localization to differences between quenched and annealed rate functions.

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.