Large Deviations for a Random Walk in Random Environment

• Published in: The Annals of probability 1994-07, Vol.22 (3), p.1381-1428
• Main Authors: Greven, Andreas, den Hollander, Frank
• Format: Article
• Language: English
• Subjects: 60F10; 60J15; 82C44; Continued fractions; Entropy; Ergodic theory; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; large deviations; Limit theorems; Markov chains; Markov processes; Mathematical methods in physics; Mathematics; Physics; Point masses; Probability and statistics; Probability theory and stochastic processes; Probability theory, stochastic processes, and statistics; Random variables; Random walk; Random walk in random environment; Sciences and techniques of general use; Statistical physics, thermodynamics, and nonlinear dynamical systems; Stochastic processes; Transition probabilities; Truncation
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1176988607

Let ω = (px)x∈Zbe an i.i.d. collection of (0, 1)-valued random variables. Given ω, let (Xn)n ≥ 0be the Markov chain on Z defined by X0= 0 and$X_{n + 1} = X_n + 1(\operatorname{resp}. X_n - 1)$with probability$p_{X_n}(\operatorname{resp}.1 - p_{X_n})$. It is shown that Xn/n satisfies a large deviation principle with a continuous rate function, that is,$\lim_{n\rightarrow\infty}\frac{1}{n}\log P_\omega(X_n = \lfloor\theta_nn\rfloor) = -I(\theta) \omega-\mathrm{a.s.}\text{for} \theta_n\rightarrow\in\lbrack -1, 1\rbrack.$First, we derive a representation of the rate function I in terms of a variational problem. Second, we solve the latter explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of I. In the recurrent case I is nonanalytic at θ = 0. In the transient case I is nonanalytic at θ = -θc, 0, θcfor some θc≥ 0, with linear pieces in between.