  We consider a random walk in a random potential on a square lattice of
arbitrary dimension. The potential is a function of an ergodic environment and
some steps of the walk. The potential can be unbounded, but it is subject to a
moment assumption whose strictness is tied to the mixing of the environment,
the best case being the i.i.d. environment. We prove that the infinite volume
quenched point-to-point free energy exists and has a variational formula in
terms of an entropy. We establish regularity properties of the point-to-point
free energy, as a function of the potential and as a function on the convex
hull of the admissible steps of the walk, and link it to the infinite volume
free energy and quenched large deviations of the endpoint of the walk. One
corollary is a quenched large deviation principle for random walk in an ergodic
random environment, with a continuous rate function.
