Slow movement of a random walk on the range of a random walk in the presence of an external field (Q389269)

The author deals with the behavior of a biased random walk moving on a random path. The latter is chosen to be the trace of a simple two-sided random walk in dimension \(d \geq 5\). The main result of the article shows that, under the annealed measure, the process undergoes a slowdown as soon as an external field, or bias, is added in one of the coordinate directions. The localisation result also implies that a higher bias makes more likely to find the random walk close to the origin. The proof is carried out through the study of the cut-points of the underlying random walk and a comparison with a one-dimensional Sinai random walk. As a byproduct, the author obtains an aging result for the position of the walker. The main result of the article points out a substantial difference between the analyzed process and biased random walk on the supercritical percolation cluster: For this second model, in fact, the slowdown effect is known to happen only above a strictly positive value of the bias (see, e.g. [\textit{N. Berger} et al., Probab. Theory Relat. Fields 126, No. 2, 221--242 (2003; Zbl 1029.60087)]).