Large Deviations and Phase Transition for Random Walks in Random   Nonnegative Potentials
  We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on $\mathbb Z^d$. We complement the analysis of \cite{Zer}, where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting. 
Markus Flury