Variational formulas and cocycle solutions for directed polymer and percolation models

Georgiou, Nicos, Rassoul-Agha, Firas and Seppäläinen, Timo (2016) Variational formulas and cocycle solutions for directed polymer and percolation models. Communications in Mathematical Physics, 346 (2). pp. 741-779. ISSN 0010-3616

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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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
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Depositing User: Nicos Georgiou
Date Deposited: 06 May 2016 11:17
Last Modified: 09 Mar 2021 09:46

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