Slowdown estimates for ballistic random walk in random environment

Abstract

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition (T ′ ). We show that for every ǫ > 0 and n large enough, the annealed probability of linear slowdown is bounded from above by exp −(log n) d−ǫ . This bound almost matches the known lower bound of exp −C(log n) d , and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability of slowdown. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the assumption of the same condition.

Effective Polynomial Ballisticity Conditions for Random Walk in Random Environment

“…and if it consecutively leaves J k boxes of scale k 1, then at least 20 (we could do significantly better here-however, since this is sufficient for our purposes, we leave it this way for the sake of simplicity) such boxes must have been left through the frontal parts of their boundaries. 1 Thus, we have that P y;! .Y j 2 B k 8j 2 f1; : : : ; J k g/ Ä J 20 k .expf c 0 k 1 N k 1 g/ 20 : This in combination with (3.21) and the Markov property applied at times that are multiples of J k supplies us with…”

“…This completes the proof of (3.18) for k. 1 Indeed, for each box of scale k 1 that .Y n / (feels and) leaves through its frontal boundary part, the position of the walk gains at least N k 2 1 units in direction l. The "most efficient" way to decrease its position in the l-direction is to leave a box of scale k 1 through its back or side boundary part, which would decrease the l-coordinate of its position by at most N k 1 C 1. Therefore, if .Y n / has not left B k through its frontal or back boundary part within J k steps, then it must have left at least J k Proof of (3.19) for k: In addition to the induction assumption that (3.18) and (3.19) hold for scale k 1, we can now assume that (3.18) holds for scale k also.…”

“…1I see, for instance, theorem 4.1 in Sznitman [16] for further details. Recently the condition .T 0 / has also been used to obtain further knowledge about large deviations for RWRE; see, e.g., Berger [1].…”

An invariance principle for a class of non-ballistic random walks in random environment

“…Then for P-a.e. ω ∈ Ω, under P 0,ω , X n · / √ n converges in law to a d-dimensional Brownian motion with diffusion matrix D −1 Λ, where D is the constant from Theorem 1.1 and Λ is given by (2).…”

Slowdown estimates for one-dimensional random walks in random environment with holding times

“…Finally, [20,21,2] provide estimates for the decay rate of the slowdown probability for random walk in random environment in higher dimensions. While it is interesting to see how the holding times affect such slowdown probabilities, our method relies on a certain renewal structure which is limited to the one dimensional setting.…”