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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.

MSC:

60K37 Processes in random environments
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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References:

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