Random walks in a random environment on a strip: a renormalization group approach

TL;DR

This paper introduces a renormalization group method for analyzing random walks in a random environment on a strip, revealing different scaling behaviors in recurrent and transient phases.

Contribution

It develops a novel real space renormalization group scheme that simplifies complex random walk models to effective one-dimensional problems.

Findings

Sinai scaling holds in the recurrent case

Displacement grows as a power of time in the transient phase

Model reduces to a nearest-neighbor walk after renormalization

Abstract

We present a real space renormalization group scheme for the problem of random walks in a random environment on a strip, which includes one-dimensional random walk in random environment with bounded non-nearest-neighbor jumps. We show that the model renormalizes to an effective one-dimensional random walk problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in the recurrent case, while in the sub-linear transient phase, the displacement grows as a power of the time.