Quenched large deviations for random walk in a random environment

We take the point of view of a particle performing random walk with bounded jumps on ℤd in a stationary and ergodic random environment. We prove the quenched large‐deviation principle (LDP) for the pair empirical measure of the so‐called environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an ansatz for the minimizer of this formula. When d = 1, we verify this ansatz and generalize the nearest‐neighbor result of Comets, Gantert, and Zeitouni to walks with bounded jumps. © 2009 Wiley Periodicals, Inc.