## Abstract

Suppose that the integers are assigned random variables {*ω*
_{i}} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {*X*
_{n}} (called a RWRE) which, when at *i*, moves one step to the right with probability *ω*
_{i}, and one step to the left with probability 1 − *ω*
_{i}. When the {*ω*
_{i}} sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for *X*
_{n}/_{n}, conditional upon the environment, with deterministic rate function.We consider in this paper large deviations, both conditioned on the environment (*quenched*) and averaged on the environment (*annealed)*, for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle.

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## Additional information

Research partially supported by CNRS, UMR7599. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.

Research supported by the Swiss National Science foundation under grant 8220-046518, and partially supported by the DFG. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.

Partially supported by a US-Israel BSF grant and by IHP.

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Comets, F., Gantert, N. & Zeitouni, O. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment.
*Probab Theory Relat Fields* **118**, 65–114 (2000). https://doi.org/10.1007/s004400000074

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DOI: https://doi.org/10.1007/s004400000074

### Mathematics Subject Classification (2000)

- 60J15
- 60F10
- 82C44
- 60J80

### Key words and phrases

- Random walk in random environment
- Large deviations