  We consider random walk in a space-time random potential, also known as
directed random polymer measures, on the planar square lattice with
nearest-neighbor steps and general i.i.d. weights on the vertices. We construct
covariant cocycles and use them to prove new results on existence,
uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer
measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove
non-existence of covariant or deterministically directed bi-infinite polymer
measures. Along the way, we prove almost sure existence of Busemann function
limits in directions where the limiting free energy has some regularity.
