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**Large deviations for random walks in a random environment.**
*(English)*
Zbl 1042.60071

The author establishes two large deviation principles with deterministic rate functions for the random walk in a stationary ergodic random environment in \(\mathbb Z^d\). The first is in the quenched setting, where the random realization of the jump distributions at each site of \(\mathbb Z^d\) is held fixed, and the second (harder) is in the annealed setting, in which probabilities are computed by averaging over realizations of the environment. The jump distributions at each site are required a.s. to have the probabilities of the unit positive and negative jumps in the coordinate directions uniformly bounded away from zero, and to give probability zero to all jumps longer than some fixed value \(C\). For the second LDP, these conditions are further strengthened, and the environment process is i.i.d.

See M. P. W. Zerner [Ann. Probab. 26, 1446–1476 (1998; Zbl 0937.60095)] for earlier results in \(\mathbb Z^d\), and F. Comets, N. Gantert and O. Zeitouni [Probab. Theory Relat. Fields 118, 65–114 (2000); erratum ibid. 125, 42–44 (2003; Zbl 0965.60098)] for the one-dimensional case.

See M. P. W. Zerner [Ann. Probab. 26, 1446–1476 (1998; Zbl 0937.60095)] for earlier results in \(\mathbb Z^d\), and F. Comets, N. Gantert and O. Zeitouni [Probab. Theory Relat. Fields 118, 65–114 (2000); erratum ibid. 125, 42–44 (2003; Zbl 0965.60098)] for the one-dimensional case.

Reviewer: A. D. Barbour (Zürich)

### MSC:

60K37 | Processes in random environments |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

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\textit{S. R. S. Varadhan}, Commun. Pure Appl. Math. 56, No. 8, 1222--1245 (2003; Zbl 1042.60071)

### References:

[1] | Comets, Probab Theory Related Fields 118 pp 65– (2000) |

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[3] | Ney, Ann Probab 15 pp 561– (1987) |

[4] | Rezakhanlou, Arch Ration Mech Anal 151 pp 277– (2000) |

[5] | Topics in random walks in random environment. Preprint. Available at: http://www.math.ethz.ch/?sznitman/preprint.shtml. · Zbl 1060.60102 |

[6] | Large deviations and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1984. |

[7] | Random walks in random environments. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 117-127. Higher Education Press, Beijing, 2002. · Zbl 1007.60102 |

[8] | Zerner, Ann Probab 26 pp 1446– (1998) |

[9] | Zerner, Ann Probab 29 pp 1716– (2001) |

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