Slow movement of a random walk on the range of a random walk in the presence of an external field

TL;DR

This paper proves that a biased random walk on the range of a simple random walk in high dimensions exhibits a slowdown effect due to an external bias, contrasting with behavior in percolation models.

Contribution

It introduces a new localization result for biased random walks on the range of simple random walks in high dimensions, linking it to Sinai's regime in random environments.

Findings

Bias causes slowdown in high-dimensional random walk range

Decomposition at cut-times relates to Sinai's regime

Contrasts with supercritical percolation behavior

Abstract

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime.