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

Rodrigo Bazaes

doi:10.1214/21-ecp426

Abstract

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on Zd, in dimensions two and higher. If d=2 or 3, we prove localization for (almost) all walks. In contrast, for d≥4, there is a phase transition for environments of the form ωe(x,e)=α(e)(1+eξ(x,e)), where {ξ(x)}x∈Zd is an i.i.d. sequence of random variables, and e represents the amount of disorder with respect to a simple random walk.

Highlights

In this paper, we deal with the notion of localization for random walks in random environment (RWRE)
We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on Zd, in dimensions two and higher
If d = 2 or 3, we prove localization for all walks

Summary

Introduction

We deal with the notion of localization for random walks in random environment (RWRE). The walk is localized if its asymptotic trajectory is confined to some region with positive probability. An i-th position environment is an element ω in the space Ω := M1(V )Zd (in general, we denote by M1(X) to the space of probability measures on X). We can define a random walk in the environment ω ∈ Ω starting at a point x ∈ Zd as the Markov chain X = (Xn)n∈N with law Px,ω that satisfies. More information about the model can be found in the references [12, 25]

Localization at the boundary

A different formulation

Main results

Alternative approaches to localization

An equivalent criterion for localization

Phase transition