Probability Theory and Related Fields

, Volume 157, Issue 3, pp 515-534

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • David A. CroydonAffiliated withDepartment of Statistics, University of Warwick Email author 


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.


Biased random walk Range of random walk Sinai’s walk Localisation Aging

Mathematics Subject Classification

60K37 60K35 60G50