## Large deviations for a random walk in random environment.(English)Zbl 0820.60054

Let $$\omega = (p_ x)_{x \in \mathbb{Z}}$$ be a sequence of i.i.d. r.v.s taking values in (0,1). Given $$\omega$$, let $$(X_ n)_{n\geq 0}$$ be a Markov chain with $$X_ 0 = 0$$ and $$X_{n + 1} = X_ n + 1$$ (resp. $$X_ n - 1$$) with probability $$p_{X_ n}$$ (resp. $$1 - p_{X_ n}$$). It is shown that $$X_ n/n$$ satisfies $\lim_{n \to \infty} {1\over n} \log P_ \omega(X_ n = [\theta_ n n]) = -I(\theta) \qquad \omega\text{-a.s. as } \theta_ n \to \theta \in [-1,1].$ The “rate function” $$I$$ is calculated explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of $$I$$. The paper uses ideas from the authors [Probab. Theory Relat. Fields 91, No. 2, 195-249 (1992; Zbl 0744.60079)] and J.-B. Baillon, Ph. Clément and the authors [J. Reine Angew. Math. 454, 181- 217 (1994; Zbl 0814.49033)].
Reviewer: M.Quine (Sydney)

### MSC:

 60G50 Sums of independent random variables; random walks 60F10 Large deviations 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

### Keywords:

random walk; random environment; large deviations

### Citations:

Zbl 0744.60079; Zbl 0814.49033
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