  We study the ergodic theory of stationary directed nearest-neighbor polymer
models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to
specifying a stationary distribution on the space of weights and correctors
that satisfy certain consistency conditions. We show that for prescribed weight
distribution and corrector mean, there is at most one stationary polymer
distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the
weights have more than two moments and the corrector mean vector is an extreme
point of the superdifferential of the limiting free energy, then the corrector
distribution is ergodic under each of the $e_1$ and $e_2$ shifts.
