Large deviation principle for random walk in a quenched random environment in the flow speed regime
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)].
scientific article; zbMATH DE number 1462557