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

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