Probability Theory and Related Fields

, Volume 158, Issue 3–4, pp 711–750 | Cite as

Quenched point-to-point free energy for random walks in random potentials

Article

Abstract

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and steps of the walk. The potential is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of entropy. We establish regularity properties of the point-to-point free energy, and link it to the infinite volume point-to-line free energy and quenched large deviations of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.

Keywords

Point-to-point Quenched Free energy Large deviation  Random walk Random environment Polymer Random potential RWRE  RWRP Directed polymer Stretched polymer Entropy Variational formula 

Mathematics Subject Classification (2010)

60F10 60K35 60K37 82D60 82B41 

Notes

Acknowledgments

F. Rassoul-Agha’s work was partially supported by NSF Grant DMS-0747758. T. Seppäläinen’s work was partially supported by NSF Grant DMS-1003651 and by the Wisconsin Alumni Research Foundation. The authors thank the anonymous referee for valuable comments that improved the presentation.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of UtahSalt Lake CityUSA
  2. 2.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA

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