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

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Abstract

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 a 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.

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Acknowledgements

C. Janjigian was partially supported by a postdoctoral grant from the Fondation Sciences Mathématiques de Paris while working at Université Paris Diderot. F. Rassoul-Agha was partially supported by National Science Foundation grants DMS-1407574 and DMS-1811090.

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  1. Mathematics Department, University of Utah, 155 South 1400 East, Salt Lake City, UT, 84109, USA

    Christopher Janjigian & Firas Rassoul-Agha

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  1. Christopher Janjigian
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  2. Firas Rassoul-Agha
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Correspondence to Firas Rassoul-Agha.

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Communicated by Eric A. Carlen.

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Janjigian, C., Rassoul-Agha, F. Uniqueness and Ergodicity of Stationary Directed Polymers on \(\mathbb {Z}^2\). J Stat Phys 179, 672–689 (2020). https://doi.org/10.1007/s10955-020-02541-z

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