Biased random walks on the interlacement set

Abstract

We study a biased random walk on the interlacement set of Z d for d ≥ 3. Although the walk is always transient, we can show, in the case d = 3, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.Résumé. Nous étudions la marche biaisée sur un entrelac aléatoire de Z d avec d ≥ 3. Nous montrons que la marche est transiente mais que, dans le cas d = 3, elle est sous-ballistique pour toutes les valeurs du biais et que ses déplacements sont inférieurs à n'importe quel polynôme.

Biased random walk on supercritical percolation: Anomalous fluctuations in the ballistic regime

“…(Precisely, due to lattice effects, the limits might only exist subsequentially; see [4,Conjecture 4.2] for some further discussion in this direction.) Other models where a similar trapping regime is expected to be found (and hence to which the results of this article might also apply) include biased random walk on supercritical Galton-Watson trees [11], in random conductances [22,24], on the interlacement set [25], and on the trace of another biased random walk [16,18]. One might also expect to see similar behaviour for random walk in transient random environments for which the tail of a suitable regeneration time distribution is of a suitable form, cf.…”

Conditional decoupling of random interlacements

“…In the work [9] the case G = I u was considered. It turned out that in dimension d = 3, for any value of β > 0, although still transient in the direction of the drift, the walk is not only sub-ballistic, but has also sub-polynomial speed, in the sense that its distance to the origin grows slower than t ε for any ε > 0.…”

“…We will not describe all the details of [9] here, but the main idea is the following. As in the case of the biased walk on the infinite percolation cluster, to prove zero speed one needs to show that the walk frequently gets caught in traps.…”

“…It is here that the conditional decoupling enters the scene: it is possible to use the main results of this paper to show that probability of having a trap in front of the particle (when it comes to an unexplored area) is not very small. As mentioned above, the detailed argument can be found in [9].…”

Biased random walk on dynamical percolation

“…Biased random walks in random media were investigated intensively over the last years, we refer to [1,2,4,6,7,8,9,11,12,13,14,15,16,17,18,20,28] for a non-exhaustive list and to [5] for a survey. The most prominent examples are biased random walks on Galton-Watson trees and biased random walks on supercritical percolation clusters, see [20] and [8,28] respectively.…”