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Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

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Abstract

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder.

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Correspondence to Timo Seppäläinen.

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Communicated by F. Toninelli

F. Rassoul-Agha and N. Georgiou were partially supported by National Science Foundation Grant DMS-0747758.

F. Rassoul-Agha was partially supported by National Science Foundation Grant DMS-1407574 and by Simons Foundation Grant 306576.

T. Seppäläinen was partially supported by National Science Foundation Grant DMS-1306777, by Simons Foundation Grant 338287, and by the Wisconsin Alumni Research Foundation.

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Georgiou, N., Rassoul-Agha, F. & Seppäläinen, T. Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models. Commun. Math. Phys. 346, 741–779 (2016). https://doi.org/10.1007/s00220-016-2613-z

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