Uniqueness and Ergodicity of Stationary Directed Polymers on $$\mathbb {Z}^2$$

@article{Janjigian2018UniquenessAE,
  title={Uniqueness and Ergodicity of Stationary Directed Polymers on \$\$\mathbb \{Z\}^2\$\$},
  author={Christopher Janjigian and Firas Rassoul-Agha},
  journal={arXiv: Probability},
  year={2018}
}
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 cocycles that satisfy certain consistency conditions. We show that for prescribed weight distribution and cocycle mean vector, there is at most one such distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the cocycle… 

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