## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 3 (1994), 1381-1428.

### Large Deviations for a Random Walk in Random Environment

Andreas Greven and Frank den Hollander

#### Abstract

Let $\omega = (p_x)_{x\in\mathbb{Z}}$ be an i.i.d. collection of (0, 1)-valued random variables. Given $\omega$, let $(X_n)_{n \geq 0}$ be the Markov chain on $\mathbb{Z}$ defined by $X_0 = 0$ 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 $X_n/n$ 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 $\theta = 0$. In the transient case $I$ is nonanalytic at $\theta = -\theta_c, 0, \theta_c$ for some $\theta_c \geq 0$, with linear pieces in between.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 3 (1994), 1381-1428.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988607

**Digital Object Identifier**

doi:10.1214/aop/1176988607

**Mathematical Reviews number (MathSciNet)**

MR1303649

**Zentralblatt MATH identifier**

0820.60054

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60F10: Large deviations 82C44: Dynamics of disordered systems (random Ising systems, etc.)

**Keywords**

Random walk in random environment large deviations

#### Citation

Greven, Andreas; den Hollander, Frank. Large Deviations for a Random Walk in Random Environment. Ann. Probab. 22 (1994), no. 3, 1381--1428. doi:10.1214/aop/1176988607. https://projecteuclid.org/euclid.aop/1176988607