Large deviation principle for random walk in a quenched random environment in the flow speed regime
type: Item
instance of: scholarly article
title: Large deviation principle for random walk in a quenched random environment in the flow speed regime
zbMATH Open document ID: 0964.60056
author: Tobias Povel
author: Ágoston Pisztora
published in: The Annals of Probability
publication date: 2001-07-02T00:00:00Z
review text: The authors consider a one-dimensional random walk in an i.i.d.\ random environment. The random environment has \`\` positive and zero drifts\'\': depending on the site where the particle is, it moves either to the left or to the right with equal probabilites (\`\` fair\'\'\ sites), or it has a drift to the right (\`\` biased\'\'\ sites). In such an environment, the random walk has almost surely a positive, deterministic speed \(v_\alpha\). For almost all environments, the probabilities that the average position \(X_n/n\) is smaller than \(v\), for some \(v < v_\alpha\), decays for \(n\) going to infinity. The order of this decay is \(\exp(-n/(\log n)^2)\), see the reviewer and \textit{O. Zeitouni} [Commun. Math. Phys. 194, No. 1, 177-190 (1998)]. The authors extend this statement and provide the rate function on \((0, v_\alpha)\), thereby proving a large deviation principle of order \(n/(\log n)^2\) for \(X_n/n\) in this regime. The main technique used in the proof of the upper bound is a coarse graining procedure where the environment is split up in blocks, and the blocks are classified to be fair or biased, according to the proportion of the fair sites in the block. A similar coarse graining technique had been used in the context of Brownian motion in a Poissonian potential by one of the authors, see \textit{T. Povel} [Ann. Probab. 25, No. 4, 1735-1773 (1997; Zbl 0911.60014)].
reviewed by: Nina Gantert
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 82C44
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1462557
zbMATH Keywords: random walk in random environment
zbMATH Keywords: large deviations
zbMATH Keywords: coarse graining
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677453
OpenAlex ID: W1536730389
cites work: Weighted sums of certain dependent random variables
cites work: Q5524074
cites work: Tail estimates for one-dimensional random walk in random environment
cites work: Quenched sub-exponential tail estimates for one-dimensional random walk in random environment
cites work: Large and moderate deviations for the local time of a recurrent Markov chain on \(\mathbb{Z}^2\)
cites work: Large deviations for a random walk in random environment
cites work: Precise large deviation estimates for a one-dimensional random walk in a random environment
cites work: Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential
cites work: Random walks in a random environment
cites work: Quenched critical large deviations for Brownian motion in a Poissonian potential
DOI: 10.1214/AOP/1022677453
Recommended article: Large deviations for a Brownian motion in a drifted Brownian potential
Recommended article: Precise large deviations for random walk in random environment
Recommended article: Precise large deviation estimates for a one-dimensional random walk in a random environment
Recommended article: Slowdown estimates and central limit theorem for random walks in random environment
Recommended article: Subexponential tail asymptotics for a random walk with randomly placed one-way nodes
Recommended article: A note on quenched moderate deviations for Sinai's random walk in random environment
description: scientific article; zbMATH DE number 1462557