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

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ( $d\ge 5$ ). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as 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. Via this approach, a corresponding aging result is also proved.