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

2019/11/15 by Bazaes, Rodrigo
#60K37 #82C41 #82D30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1911.06430

Abstract

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on ℤ^d, in dimensions two and higher. Informally, this means that the walk spends a non-trivial amount of time at some point x∈ 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≥ 4 there is a phase-transition for environments of the form ω_ε(x,e)=α(e)(1+εξ(x,e)), where \ξ(x)\_{x∈ ℤ^d} is an i.i.d. sequence of random variables, and ε 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.

Related

• Localization for Branching Random Walks in Random Environment
• Localization for random walks among random obstacles in a single Euclidean ball
• Poly-logarithmic localization for random walks among random obstacles
• Localization of a one-dimensional simple random walk among power-law renewal obstacles
• Localization on 4 sites for Vertex-reinforced random walks on \mathbb Z