Lyapunov exponents, shape theorems and large deviations for the random walk in random potential

We consider the simple random walk on ℤd evolving in a potential of independent and identically distributed random variables taking values in [0,+∞]. We give optimal conditions for the existence of the quenched point-to-point Lyapunov exponent, and for different versions of a shape theorem. The method of proof applies as well to first-passage percolation, and builds up on an approach of Cox and Durrett (1981). The weakest form of shape theorem holds whenever the set of sites with finite potential percolates. Under this condition, we then show the existence of the quenched point-to-hyperplane Lyapunov exponent, and give a large deviation principle for the walk under the quenched weighted measure.