We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process-level quenched large deviation principle for the empirical measure. As a corollary we obtain LDPs for types of random walk in random environment not covered by earlier results.

We introduce a random walk in random environment associated to an underlying directed polymer model in $1+1$ dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.Comment: Published at http://dx.doi.org/10.1214/14-AOP933 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. We take the point of view of a particle performing random walk with bounded jumps on 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.

Abstract:We consider the quenched and the averaged (or annealed) large deviation rate functions I q and I a for space-time and (the usual) space-only RWRE on Z d . By Jensen's inequality, I a ≤ I q . In the space-time case, when d ≥ 3 + 1, I q and I a are known to be equal on an open set containing the typical velocity ξ o . When d = 1 + 1, we prove that I q and I a are equal only at ξ o . Similarly, when d = 2 + 1, we show that I a < I q on a punctured neighborhood of ξ o . In the space-only case, we provide a class of non-nestling walks on Z d with d = 2 or 3, and prove that I q and I a are not identically equal on any open set containing ξ o whenever the walk is in that class. This is very different from the known results for non-nestling walks on Z d with d ≥ 4.