Quenched, annealed and functional large deviations for one-dimensional random walk in random environment

Comets, Francis; Gantert, Nina; Zeitouni, Ofer

doi:10.1007/s00440-002-0234-2

Cited by 59 publications

(112 citation statements)

References 0 publications

Supporting

Mentioning

108

Contrasting

Unclassified

Order By: Relevance

“…Indeed, in the latter case the deviation from the typical position is linear in time, but we have that the large deviation rate function I satisfies I (0) = 0, and the known large deviation results only tell us that slowdown probabilities decay slower than exponentially in n (see, for instance, [1]) . We mention here that in the literature one can find some results on moderate deviations for the case of recurrent RWRE (often referred to as RWRE in "Sinai's regime"), see [2,3], and also [7] for the continuous space and time version. Now, we state the results we are going to prove in this paper.…”

Section: Introduction and Resultsmentioning

confidence: 95%

“…However, one still can say that at time n the particle is "typically" at distance roughly n κ from the origin, since the weaker result lim n→∞ ln X n / ln n = κ, P-a.s., is still valid. 1 Besides the results about the location of the particle at time n, we are interested also in the first hitting times of certain regions in space. For any set A ⊂ Z, define:…”

Section: Introduction and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On slowdown and speedup of transient random walks in random environment

Fribergh

Gantert

Popov

2009

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in (0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.Comment: 43 pages, 4 figures; to appear in Probability Theory and Related Field

show abstract

Section: Introduction and Resultsmentioning

confidence: 95%

Section: Introduction and Resultsmentioning

confidence: 99%

On slowdown and speedup of transient random walks in random environment

Fribergh

Gantert

Popov

2009

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is strongly believed that the large deviations rate function is non-analytic at 0 (cf. [6,2]), this makes any moderate deviation result interesting and rather hard to predict. Let us introduce some notations.…”

Section: Moderate Deviationsmentioning

confidence: 94%

“…In [8] Hu and Shi identified the upper limit for the walk: lim sup ξ 0 n /((ln 2 n)(ln ln ln n)) = 8/π 2 σ 2 , a.s. As for the quenched probabilities of large deviations P ω [ξ 0 t ≥ tu], the present model in Sinai's regime is covered by [6,2] and the decay is exponential in t, but also different regimes are considered when Condition S is not satisfied, leading to full variety of behaviors (see [22]). Therefore, in the case of Sinai's regime, there remains a big gap corresponding to moderate deviations, in spite of partial results in [3,8].…”

Section: Moderate Deviationsmentioning

confidence: 99%

Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment

Comets¹,

Popov

2003

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

We consider a one-dimensional random walk in random environment in the Sinai's regime. Our main result is that logarithms of the transition probabilities, after a suitable rescaling, converge in distribution as time tends to infinity, to some functional of the Brownian motion. We compute the law of this functional when the initial and final points agree. Also, among other things, we estimate the probability of being at time t at distance at least z from the initial position, when z is larger than ln 2 t, but still of logarithmic order in time.

show abstract

“…In the last decade, asymptotics for large-deviation probabilities involving one-dimensional, nearest-neighbor random walk in random environment (RWRE) have been determined quite precisely; see [6,2,4,14,13,1], and [5] for an overview. Striking partial results were obtained by Sznitman and Zerner [19,21,16,17] in the more difficult setting of multidimensional RWRE.…”

Section: Introductionmentioning

confidence: 99%

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

Dembo

Gantert²,

Peres

et al. 2002

Probab Theory Relat Fields

Self Cite

View full text Add to dashboard Cite

In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X n } on a Galton-Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X n | the distance between the node X n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X n |/n under the "quenched measure" (conditional upon T), and the rate function for the same ratio under the "annealed measure" (averaging on T according to the Galton-Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X n } is a λ-biased walk on a Galton-Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a "ubiquity" lemma for Galton-Watson trees, due to Grimmett and Kesten (1984).

show abstract

Quenched, annealed and functional large deviations for one-dimensional random walk in random environment

Cited by 59 publications

References 0 publications

On slowdown and speedup of transient random walks in random environment

On slowdown and speedup of transient random walks in random environment

Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

Contact Info

Product

Resources

About