Quenched large deviations for random walk in a random environment

Abstract

We take the point of view of a particle performing random walk with bounded jumps on ${\mathbb{Z}}^d$ in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the 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.

28 pages. I added the construction of the minimizer of the variational formula for the rate function in the one-dimensional non-nearest-neighbor case, provided an explicit formula for the rate function, and indicated a qualitative difference from the nearest-neighbor case. Apart from this major revision, I also added three appendices

References in corpus (6)

• Limiting distributions and large deviations for random walks in random environments
• Transient random walks on a strip in a random environment
• Quenched large deviations for multidimensional random walk in random environment: a variational formula
• Limiting velocity of high-dimensional random walk in random environment
• On the annealed large deviation rate function for a multi-dimensional random walk in random environment
• Large deviations for random walk in a random environment

Cited by in corpus (3)

• Stochastic homogenization of viscous Hamilton-Jacobi equations and applications
• Variational formulas and disorder regimes of random walks in random potentials
• On the annealed large deviation rate function for a multi-dimensional random walk in random environment